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## Joint Variation – Formula, Examples | How to Solve Problems Involving Joint Variation?

Joint Variation definition, rules, methods and formulae are here. Check the joint variation problems and solutions to prepare for the exam. Refer to problems of direct and inverse variations and the relationship between the variables. Know the different type of variations like inverse, direct, combined and joint variation. Go through the below sections to check definition, various properties, example problems, value tables, concepts etc.

## Joint Variation – Introduction

Joint Variation refers to the scenario where the value of 1 variable depends on 2 or more and other variables that are held constant. For example, if C varies jointly as A and B, then C = ABX for which constant “X”. The joint variation will be useful to represent interactions of multiple variables at one time.

Most of the situations are complicated than the basic inverse or direct variation model. One or the other variables depends on the multiple other variables. Joint Variation is nothing but the variable depending on 2 or more variables quotient or product. To understand clearly with an example, The amount of busing candidates for each of the school trip varies with the no of candidates attending the distance from the school. The variable c (cost) varies jointly with n (number of students) and d (distance).

Joint Variation problems are very easy once you get the perfection of the lingo. These problems involve simple formulae or relationships which involves one variable which is equal to the “one” term which may be linear (with just an “x” axis), a quadratic equation (like “x²) where more than one variable (like “hr²”), and square root (like “\sqrt{4 – r^2\,}4−r2​”) etc.

## Functions of 2 or More Variables

It is very uncommon for the output variable to depend on 2 or more inputs. Most of the familiar formulas describe the several variables functions. For suppose, if the rectangle perimeter depends on the length and width. The cylinder volume depends on its height and radius. The travelled distance depends on the time and speed while travelling. The function notation of the formulas can be written as

P = f(l,w) = 2l + 2w where P is the perimeter and is a function of width and length

V = f(r,h) = Πr²h where V is the volume and is a function of radius and height

d = f(r,t) = rt where d is the distance and is a function of time and rate.

## Tables of Values

Just for the single variable functions, we use the tables to describe two-variable functions. The heading of the table shows row and column and it shows the value if two input variables and the complete table shows the values of the output variable.

You can easily make graphs in three dimensions for two-variable functions. Instead of representing graphs, we represent functions by holding two or one variable constants.

• What is Variation
• Practice Test on Ratio and Proportion

## How to Solve Joint Variation Problems?

Follow the step by step procedure provided below to solve problems involving Joint Variation and arrive at the solution easily. They are along the lines

Step 1: Write the exact equation. The problems of joint variation can be solved using the equation y =kxz. While dealing with the word problems. you should also consider using variables other than x,y and z. Use the variables which are relevant to the problem being solved. Read the problem carefully and determine the changes in the equation of joint variation such as cubes, squares or square roots.

Step 2: With the help of the information in the problem, you have to find the value of k which is called the constant of proportionality and variation.

Step 3: Rewrite the equation starting with 1 substituting the value of k and found in step 2.

Step 4: Use the equation in step 3 and the information in the problem to answer the question. While solving the word problems, remember including the units in the final answer.

## Joint Variation Problems with Solutions

The area of a triangle varies jointly as the base and the height. Area = 12m² when base = 6m and height = 4m. Find base when Area = 36m² and height = 8m?

The area of the triangle is represented with A

The base is represented with b

Height is represented with h

As given in the question,

A = 12m² when B = 6m and H = 4m

We know the equation,

A = kbh where k is the constant value

12 = k(6)(4)

Divide by 24 on both sides, we get

12/24 = k(24)/24

The value of k = 1/2

As the equation is

To find the base of the triangle of A = 36m² and H = 8m

36 = 1/2(b)(8)

Dividing both sides by 4, we get

36/4 = 4b/4

The value of base = 9m

Hence, the base of the triangle when A = 36m² and H = 8m is 9m

Wind resistance varies jointly as an object’s surface velocity and area. If the object travels at 80 miles per hour and has a surface area of 30 square feet which experiences 540 newtons wind resistance. How much fast will the car move with 40 square feet of the surface area in order to experience a wind resistance of 495 newtons?

Let w be the wind resistance

Let s be the object’s surface area

Let v be the object velocity

The object’s surface area = 80 newtons

The wind resistance = 540 newtons

The object velocity = 30

w = ksv where k is the constant

(540) = k (80) (30)

540 = k (2400)

540/2400 = k

The value of k is 9/40

To find the velocity of the car with s = 40, w = 495 newtons and k = 9/40

Substitute the values in the equation

495 = (9/40) (40) v

The velocity of a car is 55mph for which the object’s surface area is 40 and wind resistance is 495 newtons

Hence, the final solution is 55mph

For the given interest, SI (simple interest) varies jointly as principal and time. If 2,500 Rs left in an account for 5 years, then the interest of 625 Rs. How much interest would be earned, if you deposit 7,000 Rs for 9 years?

Let i be the interest

Let p be the principal

Let t be the time

The interest is 625 Rs

The principal is 2500

The time is 5 hours

i = kpt where k is the constant

Substituting the values in the equation,

(625) = k(2500)(5)

625 = k(12,500)

Dividing 12,500 on both the sides

625/12,500 = k (12,500)/12,500

The value of k = 1/20

To find the interest where the deposit is 7000Rs for 9 years, use the equation

i = (1/20) (7000) (9)

i = (350) (9)

Therefore, the interest is 3,150 Rs, if you deposit 7,000 Rs for 9 years

Thus, the final solution is Rs. 3,150

The volume of a pyramid varies jointly as its height and the area of the base. A pyramid with a height of 21 feet and a base with an area of 24 square feet has a volume of 168 cubic feet. Find the volume of a pyramid with a height of 18 feet and a base with an area of 42 square feet?

Let v be the volume of a pyramid

Let h be the height of a pyramid

Let a be the area of a pyramid

The volume v = 168 cubic feet

The height h = 21 feet

The area a = 24 square feet

V = Kha where K is the constant,

168 = k(21)(24)

168 = k(504)

Divide 504 on both sides

168/504 = k(504)/504

The value of k = 1/3

To find the volume of a pyramid with a height of 18 feet and a base with an area of 42 square feet

h = 18 feet

a = 42 square feet

V = (1/3) (18) (42)

V = (6) (42)

V = 252 ft³

The volume of the pyramid = 252 ft³ which has a height of 18 feet and a base with an area of 42 square feet

Therefore, the final solution is 252 ft³

The amount of oil used by a ship travelling at a uniform speed varies jointly with the distance and the square of the speed. If the ship uses 200 barrels of oil in travelling 200 miles at 36 miles per hour, determine how many barrels of oil are used when the ship travels 360 miles at 18 miles per hour?

No of barrels of oil = 200

The distance at which the oil is travelling = 200 miles

The distance at which the ship is travelling = 36 miles per hour

A = kds² where k is constant

200 = k.200.(36)²

Dividing both sides by 200

200/200 = k.200.(36)²/200

1 = k.(36)²

The value of k is 1/1296

To find the no of barrels when the ship travels 360 miles at 18 miles per hour

A = 1/1296 * 360 * 18²

Therefore, 90 barrels of oil is used when the ship travels 360 miles at 18 miles per hour

Thus, the final solution is 90 barrels

• Joint Variation

If more than two variables are related directly or one variable changes with the change product of two or more variables it is called as joint variation .

If X is in joint variation with Y and Z, it can be symbolically written as X α YZ. If Y is constant also then X is in direct variation with Z. So for joint variation two or more variables are separately in direct variation. So joint variation is similar to direct variation but the variables for joint variation are more than two.

Equation for a joint variation is X = KYZ where K is constant.

One variable quantity is said to vary jointly as a number of other variable quantities, when it varies directly as their product. If the variable A varies directly as the product of the variables B, C and D, i.e., if.A ∝ BCD or A = kBCD (k = constant ), then A varies jointly as B, C and D.

For solving a problems related to joint variation first we need to build the correct equation by adding a constant and relate the variables. After that we need determine the value of the constant. Then substitute the value of the constant in the equation and by putting the values of variables for required situation we determine the answer.

We know, area of a triangle = ½ × base × altitude. Since ½ is a constant, hence area of a triangle varies jointly as its base and altitude.  A is said to vary directly as B and inversely as C if A ∝ B ∙ $$\frac{1}{C}$$  or A = m ∙ B ∙ $$\frac{1}{C}$$  (m = constant of variation) i.e., if A varies jointly as B and $$\frac{1}{C}$$ .

If x men take y days to plough z acres of land, then x varies directly as z and inversely as y.

1.  The variable x is in joint variation with y and z. When the values of y and z are 4 and 6, x is 16. What is the value of x when y = 8 and z =12?

The equation for the given problem of joint variation is

x = Kyz where K is the constant.

For the given data

16 = K  ×  4  ×  6

or, K = $$\frac{4}{6}$$ .

So substituting the value of K the equation becomes

x =  $$\frac{4yz}{6}$$

Now for the required condition

x = $$\frac{4 × 8 × 12}{6}$$

= 64

Hence the value of x will be 64.

2.  A is in joint variation with B and square of C. When A = 144, B = 4 and C = 3. Then what is the value of A when B = 6 and C = 4?

From the given problem equation for the joint variation is

From the given data value of the constant K is

K =   $$\frac{BC^{2}}{A}$$

K = $$\frac{4 × 3^{2}}{144}$$

=  $$\frac{36}{144}$$

= $$\frac{1}{4}$$ .

Substituting the value of K in the equation

A =  $$\frac{BC^{2}}{4}$$

A = $$\frac{6 × 4^{2}}{4}$$

= 24

3. The area of a triangle is jointly related to the height and the base of the triangle. If the base is increased 10% and the height is decreased by 10%, what will be the percentage change of the area?

We know the area of triangle is half the product of base and height. So the joint variation equation for area of triangle is A = $$\frac{bh}{2}$$  where A is the area, b is the base and h is the height.

Here $$\frac{1}{2}$$  is the constant for the equation.

Base is increased by 10%, so it will be b x $$\frac{110}{100}$$  = $$\frac{11b}{10}$$ .

Height is decreased by 10%, so it will be h x $$\frac{90}{100}$$  = $$\frac{9h}{10}$$ .

So the new area after the changes of base and height is

$$\frac{\frac{11b}{10} \times \frac{9h}{10}}{2}$$

= ($$\frac{99}{100}$$ )$$\frac{bh}{2}$$  = $$\frac{99}{100}$$ A.

So the area of the triangle is decreased by 1%.

4. A rectangle’s length is 6 m and width is 4 m. If length is doubled and width is halved, how much the perimeter will increase or decrease?

Formula for the perimeter of rectangle is P = 2(l + w) where P is perimeter, l is length and w is width.

This is joint variation equation where 2 is constant.

So P = 2(6 + 4) = 20 m

If length is doubled, it will become 2l.

And width is halved, so it will become $$\frac{w}{2}$$ .

So the new perimeter will be P = 2(2l +  $$\frac{w}{2}$$ ) = 2(2 x 6 +  $$\frac{4}{2}$$ ) = 28 m.

So the perimeter will increase by (28 - 20) = 8 m.

●   Variation

• What is Variation?
• Direct Variation
• Inverse or Indirect Variation
• Theorem of Joint Variation
• Worked out Examples on Variation
• Problems on Variation

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Chapter 2: Linear Equations

## 2.7 Variation Word Problems

Direct variation problems.

There are many mathematical relations that occur in life. For instance, a flat commission salaried salesperson earns a percentage of their sales, where the more they sell equates to the wage they earn. An example of this would be an employee whose wage is 5% of the sales they make. This is a direct or a linear variation, which, in an equation, would look like:

$\text{Wage }(x)=5\%\text{ Commission }(k)\text{ of Sales Completed }(y)$

$x=ky$

A historical example of direct variation can be found in the changing measurement of pi, which has been symbolized using the Greek letter π since the mid 18th century. Variations of historical π calculations are Babylonian $\left(\dfrac{25}{8}\right),$ Egyptian $\left(\dfrac{16}{9}\right)^2,$ and Indian $\left(\dfrac{339}{108}\text{ and }10^{\frac{1}{2}}\right).$ In the 5th century, Chinese mathematician Zu Chongzhi calculated the value of π to seven decimal places (3.1415926), representing the most accurate value of π for over 1000 years.

Pi is found by taking any circle and dividing the circumference of the circle by the diameter, which will always give the same value: 3.14159265358979323846264338327950288419716… (42 decimal places). Using an infinite-series exact equation has allowed computers to calculate π to 10 13 decimals.

$\begin{array}{c} \text{Circumference }(c)=\pi \text{ times the diameter }(d) \\ \\ \text{or} \\ \\ c=\pi d \end{array}$

All direct variation relationships are verbalized in written problems as a direct variation or as directly proportional and take the form of straight line relationships. Examples of direct variation or directly proportional equations are:

• $x$ varies directly as $y$
• $x$ varies as $y$
• $x$ varies directly proportional to $y$
• $x$ is proportional to $y$
• $x$ varies directly as the square of $y$
• $x$ varies as $y$ squared
• $x$ is proportional to the square of $y$
• $x$ varies directly as the cube of $y$
• $x$ varies as $y$ cubed
• $x$ is proportional to the cube of $y$
• $x$ varies directly as the square root of $y$
• $x$ varies as the root of $y$
• $x$ is proportional to the square root of $y$

Example 2.7.1

Find the variation equation described as follows:

The surface area of a square surface $(A)$ is directly proportional to the square of either side $(x).$

$\begin{array}{c} \text{Area }(A) =\text{ constant }(k)\text{ times side}^2\text{ } (x^2) \\ \\ \text{or} \\ \\ A=kx^2 \end{array}$

Example 2.7.2

When looking at two buildings at the same time, the length of the buildings’ shadows $(s)$ varies directly as their height $(h).$ If a 5-story building has a 20 m long shadow, how many stories high would a building that has a 32 m long shadow be?

The equation that describes this variation is:

$h=kx$

Breaking the data up into the first and second parts gives:

$\begin{array}{ll} \begin{array}{rrl} \\ &&\textbf{1st Data} \\ s&=&20\text{ m} \\ h&=&5\text{ stories} \\ k&=&\text{find 1st} \\ \\ &&\text{Find }k\text{:} \\ h&=&kx \\ 5\text{ stories}&=&k\text{ (20 m)} \\ k&=&5\text{ stories/20 m}\\ k&=&0.25\text{ story/m} \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ s&=&\text{32 m} \\ h&=&\text{find 2nd} \\ k&=&0.25\text{ story/m} \\ \\ &&\text{Find }h\text{:} \\ h&=&kx \\ h&=&(0.25\text{ story/m})(32\text{ m}) \\ h&=&8\text{ stories} \end{array} \end{array}$

## Inverse Variation Problems

Inverse variation problems are reciprocal relationships. In these types of problems, the product of two or more variables is equal to a constant. An example of this comes from the relationship of the pressure $(P)$ and the volume $(V)$ of a gas, called Boyle’s Law (1662). This law is written as:

$\begin{array}{c} \text{Pressure }(P)\text{ times Volume }(V)=\text{ constant} \\ \\ \text{ or } \\ \\ PV=k \end{array}$

Written as an inverse variation problem, it can be said that the pressure of an ideal gas varies as the inverse of the volume or varies inversely as the volume. Expressed this way, the equation can be written as:

$P=\dfrac{k}{V}$

Another example is the historically famous inverse square laws. Examples of this are the force of gravity $(F_{\text{g}}),$ electrostatic force $(F_{\text{el}}),$ and the intensity of light $(I).$ In all of these measures of force and light intensity, as you move away from the source, the intensity or strength decreases as the square of the distance.

In equation form, these look like:

$F_{\text{g}}=\dfrac{k}{d^2}\hspace{0.25in} F_{\text{el}}=\dfrac{k}{d^2}\hspace{0.25in} I=\dfrac{k}{d^2}$

These equations would be verbalized as:

• The force of gravity $(F_{\text{g}})$ varies inversely as the square of the distance.
• Electrostatic force $(F_{\text{el}})$ varies inversely as the square of the distance.
• The intensity of a light source $(I)$ varies inversely as the square of the distance.

All inverse variation relationship are verbalized in written problems as inverse variations or as inversely proportional. Examples of inverse variation or inversely proportional equations are:

• $x$ varies inversely as $y$
• $x$ varies as the inverse of $y$
• $x$ varies inversely proportional to $y$
• $x$ is inversely proportional to $y$
• $x$ varies inversely as the square of $y$
• $x$ varies inversely as $y$ squared
• $x$ is inversely proportional to the square of $y$
• $x$ varies inversely as the cube of $y$
• $x$ varies inversely as $y$ cubed
• $x$ is inversely proportional to the cube of $y$
• $x$ varies inversely as the square root of $y$
• $x$ varies as the inverse root of $y$
• $x$ is inversely proportional to the square root of $y$

Example 2.7.3

The force experienced by a magnetic field $(F_{\text{b}})$ is inversely proportional to the square of the distance from the source $(d_{\text{s}}).$

$F_{\text{b}} = \dfrac{k}{{d_{\text{s}}}^2}$

Example 2.7.4

The time $(t)$ it takes to travel from North Vancouver to Hope varies inversely as the speed $(v)$ at which one travels. If it takes 1.5 hours to travel this distance at an average speed of 120 km/h, find the constant $k$ and the amount of time it would take to drive back if you were only able to travel at 60 km/h due to an engine problem.

$t=\dfrac{k}{v}$

$\begin{array}{ll} \begin{array}{rrl} &&\textbf{1st Data} \\ v&=&120\text{ km/h} \\ t&=&1.5\text{ h} \\ k&=&\text{find 1st} \\ \\ &&\text{Find }k\text{:} \\ k&=&tv \\ k&=&(1.5\text{ h})(120\text{ km/h}) \\ k&=&180\text{ km} \end{array} & \hspace{0.5in} \begin{array}{rrl} \\ \\ \\ &&\textbf{2nd Data} \\ v&=&60\text{ km/h} \\ t&=&\text{find 2nd} \\ k&=&180\text{ km} \\ \\ &&\text{Find }t\text{:} \\ t&=&\dfrac{k}{v} \\ \\ t&=&\dfrac{180\text{ km}}{60\text{ km/h}} \\ \\ t&=&3\text{ h} \end{array} \end{array}$

## Joint or Combined Variation Problems

In real life, variation problems are not restricted to single variables. Instead, functions are generally a combination of multiple factors. For instance, the physics equation quantifying the gravitational force of attraction between two bodies is:

$F_{\text{g}}=\dfrac{Gm_1m_2}{d^2}$

• $F_{\text{g}}$ stands for the gravitational force of attraction
• $G$ is Newton’s constant, which would be represented by $k$ in a standard variation problem
• $m_1$ and $m_2$ are the masses of the two bodies
• $d^2$ is the distance between the centres of both bodies

To write this out as a variation problem, first state that the force of gravitational attraction $(F_{\text{g}})$ between two bodies is directly proportional to the product of the two masses $(m_1, m_2)$ and inversely proportional to the square of the distance $(d)$ separating the two masses. From this information, the necessary equation can be derived. All joint variation relationships are verbalized in written problems as a combination of direct and inverse variation relationships, and care must be taken to correctly identify which variables are related in what relationship.

Example 2.7.5

The force of electrical attraction $(F_{\text{el}})$ between two statically charged bodies is directly proportional to the product of the charges on each of the two objects $(q_1, q_2)$ and inversely proportional to the square of the distance $(d)$ separating these two charged bodies.

$F_{\text{el}}=\dfrac{kq_1q_2}{d^2}$

Solving these combined or joint variation problems is the same as solving simpler variation problems.

First, decide what equation the variation represents. Second, break up the data into the first data given—which is used to find $k$—and then the second data, which is used to solve the problem given. Consider the following joint variation problem.

Example 2.7.6

$y$ varies jointly with $m$ and $n$ and inversely with the square of $d$. If $y = 12$ when $m = 3$, $n = 8$, and $d = 2,$ find the constant $k$, then use $k$ to find $y$ when $m=-3$, $n = 18$, and $d = 3$.

$y=\dfrac{kmn}{d^2}$

$\begin{array}{ll} \begin{array}{rrl} \\ \\ \\ && \textbf{1st Data} \\ y&=&12 \\ m&=&3 \\ n&=&8 \\ d&=&2 \\ k&=&\text{find 1st} \\ \\ &&\text{Find }k\text{:} \\ y&=&\dfrac{kmn}{d^2} \\ \\ 12&=&\dfrac{k(3)(8)}{(2)^2} \\ \\ k&=&\dfrac{12(2)^2}{(3)(8)} \\ \\ k&=& 2 \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ y&=&\text{find 2nd} \\ m&=&-3 \\ n&=&18 \\ d&=&3 \\ k&=&2 \\ \\ &&\text{Find }y\text{:} \\ y&=&\dfrac{kmn}{d^2} \\ \\ y&=&\dfrac{(2)(-3)(18)}{(3)^2} \\ \\ y&=&12 \end{array} \end{array}$

For questions 1 to 12, write the formula defining the variation, including the constant of variation $(k).$

• $x$ is jointly proportional to $y$ and $z$
• $x$ varies jointly as $z$ and $y$
• $x$ is jointly proportional with the square of $y$ and the square root of $z$
• $x$ is inversely proportional to $y$ to the sixth power
• $x$ is jointly proportional with the cube of $y$ and inversely to the square root of $z$
• $x$ is inversely proportional with the square of $y$ and the square root of $z$
• $x$ varies jointly as $z$ and $y$ and is inversely proportional to the cube of $p$
• $x$ is inversely proportional to the cube of $y$ and square of $z$

For questions 13 to 22, find the formula defining the variation and the constant of variation $(k).$

• If $A$ varies directly as $B,$ find $k$ when $A=15$ and $B=5.$
• If $P$ is jointly proportional to $Q$ and $R,$ find $k$ when $P=12, Q=8$ and $R=3.$
• If $A$ varies inversely as $B,$ find $k$ when $A=7$ and $B=4.$
• If $A$ varies directly as the square of $B,$ find $k$ when $A=6$ and $B=3.$
• If $C$ varies jointly as $A$ and $B,$ find $k$ when $C=24, A=3,$ and $B=2.$
• If $Y$ is inversely proportional to the cube of $X,$ find $k$ when $Y=54$ and $X=3.$
• If $X$ is directly proportional to $Y,$ find $k$ when $X=12$ and $Y=8.$
• If $A$ is jointly proportional with the square of $B$ and the square root of $C,$ find $k$ when $A=25, B=5$ and $C=9.$
• If $y$ varies jointly with $m$ and the square of $n$ and inversely with $d,$ find $k$ when $y=10, m=4, n=5,$ and $d=6.$
• If $P$ varies directly as $T$ and inversely as $V,$ find $k$ when $P=10, T=250,$ and $V=400.$

For questions 23 to 37, solve each variation word problem.

• The electrical current $I$ (in amperes, A) varies directly as the voltage $(V)$ in a simple circuit. If the current is 5 A when the source voltage is 15 V, what is the current when the source voltage is 25 V?
• The current $I$ in an electrical conductor varies inversely as the resistance $R$ (in ohms, Ω) of the conductor. If the current is 12 A when the resistance is 240 Ω, what is the current when the resistance is 540 Ω?
• Hooke’s law states that the distance $(d_s)$ that a spring is stretched supporting a suspended object varies directly as the mass of the object $(m).$ If the distance stretched is 18 cm when the suspended mass is 3 kg, what is the distance when the suspended mass is 5 kg?
• The volume $(V)$ of an ideal gas at a constant temperature varies inversely as the pressure $(P)$ exerted on it. If the volume of a gas is 200 cm 3 under a pressure of 32 kg/cm 2 , what will be its volume under a pressure of 40 kg/cm 2 ?
• The number of aluminum cans $(c)$ used each year varies directly as the number of people $(p)$ using the cans. If 250 people use 60,000 cans in one year, how many cans are used each year in a city that has a population of 1,000,000?
• The time $(t)$ required to do a masonry job varies inversely as the number of bricklayers $(b).$ If it takes 5 hours for 7 bricklayers to build a park wall, how much time should it take 10 bricklayers to complete the same job?
• The wavelength of a radio signal (λ) varies inversely as its frequency $(f).$ A wave with a frequency of 1200 kilohertz has a length of 250 metres. What is the wavelength of a radio signal having a frequency of 60 kilohertz?
• The number of kilograms of water $(w)$ in a human body is proportional to the mass of the body $(m).$ If a 96 kg person contains 64 kg of water, how many kilograms of water are in a 60 kg person?
• The time $(t)$ required to drive a fixed distance $(d)$ varies inversely as the speed $(v).$ If it takes 5 hours at a speed of 80 km/h to drive a fixed distance, what speed is required to do the same trip in 4.2 hours?
• The volume $(V)$ of a cone varies jointly as its height $(h)$ and the square of its radius $(r).$ If a cone with a height of 8 centimetres and a radius of 2 centimetres has a volume of 33.5 cm 3 , what is the volume of a cone with a height of 6 centimetres and a radius of 4 centimetres?
• The centripetal force $(F_{\text{c}})$ acting on an object varies as the square of the speed $(v)$ and inversely to the radius $(r)$ of its path. If the centripetal force is 100 N when the object is travelling at 10 m/s in a path or radius of 0.5 m, what is the centripetal force when the object’s speed increases to 25 m/s and the path is now 1.0 m?
• The maximum load $(L_{\text{max}})$ that a cylindrical column with a circular cross section can hold varies directly as the fourth power of the diameter $(d)$ and inversely as the square of the height $(h).$ If an 8.0 m column that is 2.0 m in diameter will support 64 tonnes, how many tonnes can be supported by a column 12.0 m high and 3.0 m in diameter?
• The volume $(V)$ of gas varies directly as the temperature $(T)$ and inversely as the pressure $(P).$ If the volume is 225 cc when the temperature is 300 K and the pressure is 100 N/cm 2 , what is the volume when the temperature drops to 270 K and the pressure is 150 N/cm 2 ?
• The electrical resistance $(R)$ of a wire varies directly as its length $(l)$ and inversely as the square of its diameter $(d).$ A wire with a length of 5.0 m and a diameter of 0.25 cm has a resistance of 20 Ω. Find the electrical resistance in a 10.0 m long wire having twice the diameter.
• The volume of wood in a tree $(V)$ varies directly as the height $(h)$ and the diameter $(d).$ If the volume of a tree is 377 m 3 when the height is 30 m and the diameter is 2.0 m, what is the height of a tree having a volume of 225 m 3 and a diameter of 1.75 m?

## Joint or Combined Variation

These lessons help Algebra students learn about joint or combined variation.

Related Pages: Proportions Joint Variation Word Problems Direct Variation Inverse Variation More Algebra Lessons

The following figure shows Joint Variation. Scroll down the page for more examples and solutions of Joint and Combine Variations.

## What is Joint Variation or Combined Variation?

Joint Variation or Combined Variation is when one quantity varies directly as the product of at least two other quantities.

For example: y = kxz y varies jointly as x and z, when there is some nonzero constant k

Joint Variation Examples

Example: Suppose y varies jointly as x and z. What is y when x = 2 and z = 3, if y = 20 when x = 4 and z = 3?

Example: z varies jointly with x and y. When x = 3, y = 8, z = 6. Find z, when x = 6 and y = 4.

Joint Variation Application

Example: The energy that an item possesses due to its motion is called kinetic energy. The kinetic energy of an object (which is measured in joules) varies jointly with the mass of the object and the square of its velocity. If the kinetic energy of a 3 kg ball traveling 12 m/s is 216 Joules, how is the mass of a ball that generates 250 Joules of energy when traveling at 10 m/s?

Distinguish between Direct, Inverse and Joint Variation

Example: Determine whether the data in the table is an example of direct, inverse or joint variation. Then, identify the equation that represents the relationship.

Combined Variation

In Algebra, sometimes we have functions that vary in more than one element. When this happens, we say that the functions have joint variation or combined variation. Joint variation is direct variation to more than one variable (for example, d = (r)(t)). With combined variation, we have both direct variation and indirect variation.

How to set up and solve combined variation problems?

Example: Suppose y varies jointly with x and z. When y = 20, x = 6 and z = 10. Find y when x = 8 and z =15.

Lesson on combining direct and inverse or joint and inverse variation

Example: y varies directly as x and inversely as the square of z, and when x = 32, y = 6 and z = 4. Find x when y = 10 and z = 3.

How to solve problems involving joint and combined variation?

If t varies jointly with u and the square of v, and t is 1152 when u is 8 and v is 4, find t when v is 5 and u is 5.

The amount of oil used by a ship traveling at a uniform speed varies jointly with the distance and the square of the speed. If the ship uses 200 barrels of oil in traveling 200 miles at 36 miles per hour, determine how many barrels of oil are used when the ship travels 360 miles at 18 miles per hour.

Designer Dolls found that its number of Dress-Up Dolls sold, N, varies directly with their advertising budget, A, and inversely proportional with the price of each doll, P. When $54,00 was spent on advertising and the price of the doll is$90, then 9,600 units are sold. Determine the number of dolls sold if the amount of advertising budget is increased to $144,000. Example: y varies jointly as x and z and inversely as w, and y = 3/2, when x = 2, z =3 and w = 4. Find the equation of variation. We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. ## Direct, Inverse, Joint and Combined Variation When you start studying algebra, you will also study how two (or more) variables can relate to each other specifically. The cases you’ll study are: • Direct Variation , where one variable is a constant multiple of another. For example, the number of dollars I make varies directly (or varies proportionally ) to the number of hours I work. Or, the perimeter of a square varies directly with the length of a side of the square. • Inverse or Indirect Variation , where when one of the variables increases, the other one decreases (their product is constant). For example, the temperature in my house varies indirectly (same or inversely ) with the amount of time the air conditioning is running. Or, the number of people I invite to my bowling party varies inversely with the number of games they might get to play (or you can say is proportional to the inverse of ). • Joint Variation , where at least two variables are related directly. For example, the area of a triangle is jointly related to both its height and base. • Combined Variation , which involves a combination of direct or joint variation, and indirect variation. For example, the average number of phone calls per day between two cities has found to be jointly proportional to the populations of the cities, and inversely proportional to the square of the distance between the two cities. • Partial (Direct) Variation , where two variables are related by a formula, such as the formula for a straight line (with a non-zero$ y$-intercept). For example, the total cost of my phone bill consists of a fixed cost per month, and also a charge per minute. Note : Just because two variables have a direct relationship, the relationship may not necessarily be a causal relationship (causation) , meaning one variable directly affects the other. There may be another variable that affects both of the variables. For example, there may be a correlation between the number of people buying ice cream and the number of people buying shorts. People buying ice cream do not cause people to buy shorts, but most likely warm weather outside is causing both to happen. Here is a table for the types of variation we’ll be discussing: ## Direct or Proportional Variation When two variables are related directly, the ratio of their values is always the same. If$ k$, the constant ratio is positive, the variables go up and down in the same direction. (If$ k$is negative, as one variable goes up, the other goes down; this is still considered a direct variation, but is not seen often in these problems.) Note that$ k\ne 0$. Think of linear direct variation as a “$ y=mx$” line, where the ratio of$ y$to$ x$is the slope ($ m$). With direct variation, the$ y$-intercept is always 0 (zero); this is how it’s defined. Direct variation problems are typically written: →$ \boldsymbol {y=kx}$, where$ k$is the ratio of$ y$to$ x$(which is the same as the slope or rate ). Some problems will ask for that$ k$value (which is called the constant ratio , constant of variation or constant of proportionality – it’s like a slope!); others will just give you 3 out of the 4 values for$ x$and$ y$and you can simply set up a ratio to find the other value. I’m thinking the$ k$comes from the word “constant” in another language. Remember the example of making$10 an hour at the mall ($y=10x$)? This is an example of  direct variation, since the ratio of how much you make to how many hours you work is always constant.

We can also set up direct variation problems in a ratio , as long as we have the same variable in either the top or bottom of the ratio, or on the same side . This will look like the following. Don’t let this scare you; the subscripts just refer to either the first set of variables $({{x}_{1}},{{y}_{1}})$, or the second $({{x}_{2}},{{y}_{2}})$:   $\displaystyle \frac{{{{y}_{1}}}}{{{{x}_{1}}}}\,\,=\,\,\frac{{{{y}_{2}}}}{{{{x}_{2}}}}$.

Notes: Partial Variation   (see below), or “varies partly” means that there is an extra fixed constant, so we’ll have an equation like $y=mx+b$, which is our typical linear equation. Also, I’m assuming in these examples that direct variation is linear ; sometime I see it where it’s not, like in a Direct Square Variation where $y=k{{x}^{2}}$. There is a word problem example of this here .

Direct Variation Word Problem:

We can solve the following direct variation problem in one of two ways, as shown. We do these methods when we are given any three of the four values for $x$ and $y$.

It’s really that easy. Can you see why the proportion method can be the preferred method, unless you are asked to find the $k$ constant in the formula?

Again, if the problem asks for the  equation that models this situation , it would be “$y=10x$”.

Here’s another; let’s use the proportion method :

See how similar these types of problems are to the Proportions problems we did earlier?

Direct Square Variation Word Problem:

Again, a Direct Square Variation is when $y$ is proportional to the square of $x$, or $y=k{{x}^{2}}$. Let’s work a word problem with this type of variation and show both the formula and proportion methods:

## Inverse or Indirect Variation

Inverse  or Indirect  Variation refers to relationships of two variables that go in the opposite direction (their product is a constant, $k$). Let’s suppose you are comparing how fast you are driving (average speed) to how fast you get to your school. You might have measured the following speeds and times:

(Note that $\approx$ means “approximately equal to”).

Do you see how when the $x$ variable goes up, the $y$ goes down, and when you multiply the $x$ with the $y$, we always get the same number? (Note that this is different than a negative slope, or negative $k$ value, since with a negative slope, we can’t multiply the $x$’s and $y$’s to get the same number).

The formula for inverse or indirect variation is:    →    $\displaystyle \boldsymbol{y=\frac{k}{x}}$  or  $\boldsymbol{xy=k}$, where $k$ is always the same number.

(Note that you could also have an Indirect Square Variation or Inverse Square Variation , like we saw above for a Direct Variation. This would be of the form $\displaystyle y=\frac{k}{{{{x}^{2}}}}\text{ or }{{x}^{2}}y=k$.)

Here is a sample graph for inverse or indirect variation. This is actually a type of Rational Function  (function with a variable in the denominator) that we will talk about in the Rational Functions, Equations and Inequalities section .

Inverse Variation Word Problem:

We might have a problem like this; we can solve this problem in one of two ways, as shown. We do these methods when we are given any three of the four values for $x$ and $y$:

Here’s another; let’s use the product method:

“Work” Inverse Proportion Word Problem:

Here’s a more advanced problem that uses inverse proportions in a  “work” word problem ; we’ll see more “work problems”  here in the  Systems of Linear Equations Section  and  here in the  Rational Functions and Equations section .

## Recognizing Direct or Indirect Variation

You might be asked to look at functions  (equations or points that compare $x$’s to unique $y$’s   – we’ll discuss later in the Algebraic Functions section) and determine if they are direct, inverse, or neither:

## Joint Variation and Combined Variation

Joint variation is just like direct variation, but involves more than one other variable. All the variables are directly proportional, taken one at a time. Let’s set this up like we did with direct variation, find the $k$, and then solve for $y$; we need to use the Formula Method:

Joint Variation Word Problem:

We know the equation for the area of a triangle is $\displaystyle A=\frac{1}{2}bh$ ($b=$ base and $h=$ height), so we can think of the area having a joint variation with $b$ and $h$, with $\displaystyle k=\frac{1}{2}$. Let’s do an area problem, where we wouldn’t even have to know the value for $k$:

Another Joint Variation Word Problem:

## Combined Variation

Combined variation involves a combination of direct or joint variation, and indirect variation. Since these equations are a little more complicated, you probably want to plug in all the variables, solve for $k$, and then solve back to get what’s missing. Let’s try a problem:

Combined Variation Word Problem:

Here’s another; this one looks really tough, but it’s really not that bad if you take it one step at a time:

Combined Variation Word Problem:

One word of caution : I found a variation problem in an SAT book that stated something like this: “ If $x$ varies inversely with $y$ and varies directly with $z$, and if $y$ and $z$ are both 12 when $x=3$, what is the value of $y+z$ when $x=5$ ”. I found that I had to solve it setting up two variation equations with two different $k$ ‘s  (otherwise you can’t really get an answer). So watch the wording of the problems.  🙁 Here is how I did this problem:

## Partial Variation

You don’t hear about  Partial Variation  or something being  partly varied  or  part varied very often, but it means that two variables are related by the sum of two or more variables (one of which may be a constant). An example of part variation is the relationship modeled by an equation of a line that doesn’t go through the origin. Here are a few examples:

We’re doing really difficult problems now – but see how, if you know the rules, they really aren’t bad at all?

Learn these rules, and practice, practice, practice!

For Practice : Use the Mathway  widget below to try a Variation  problem. Click on Submit (the blue arrow to the right of the problem) and click on Find the Constant of Variation  to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on Tap to view steps , or Click Here , you can register at Mathway for a free trial , and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to Introduction to the Graphing Display Calculator (GDC) . I’m proud of you for getting this far!

## JOINT VARIATION WORD PROBLEMS

Problem 1 :

z varies directly with the sum of squares of x and y. z = 5 when x = 3 and y = 4. Find the value of z when x = 2 and y = 4.

Since z varies directly with the sum of squares of x and y,

z ∝ x 2 +  x 2

z = k(x 2  + y 2 ) ----(1)

Substitute z = 5, x = 3 and y = 4 to find the value k.

5 = k(3 2  + 4 2 )

5 = k(9 + 16)

Divide both sides by 25.

Substitute k = 1/5 in (1).

z = (1/5)(x 2  + y 2 )

Substitute  x = 2, y = 4 and evaluate z.

z = (1/5)( (2 2  + 4 2 )

z = (1/5)( (4   + 16)

z = (1/5)( (20)

Problem 2 :

M varies directly with the square of d and inversely with the square root of x. M = 24 when d = 4 and x = 9. Find the value of M when d = 5 and x = 4.

Since m varies directly with the square of d and inversely with the square root of x

M ∝ d 2 √ x

M = kd 2 √ x ----(1)

Substitute M = 24, d = 4 and x = 9 to find the value k.

24 = k4 2 √9

24 = k(16)(3)

Divide both sides by 48.

Substitute k = 1/2 in (1).

M = (1/2)(d 2 √ x )

Substitute d  = 5, x = 4 and evaluate M.

M = (1/2) (5 2 √4 )

M = (1/2)( (25)(2)

Problem 3 :

Square of T varies directly with the cube of a and inversely with the square of d. T = 2 when a = 2 and d = 4. Find the value of s quare of T when a = 4 and d = 2

Since square of T varies directly with the cube of a and inversely with the square of d

T 2 ∝ a 3 d 2

T 2  = ka 3 d 2  ----(1)

Substitute T = 2, a = 2 and d = 4 to find the value k.

2 2  = k2 3 4 2

4 = k(4)(16)

Divide both sides by 64.

Substitute k = 1/16 in (1).

T 2  = (1/16)a 3 d 2

Substitute a = 4, d = 2 and evaluate T 2 .

T 2  = (1/16)(4 3 )(2 2 )

T 2  = (1/16)(64)(4)

T 2  = 16

Problem 4 :

The area of a rectangle varies directly with its length and square of its width. When the length is 5 cm and width is 4 cm, the area is 160 cm 2 . Find the area of the rectangle when the length is 7 cm and the width is 3 cm.

Let A represent the area of the rectangle, l represent the length and w represent width.

Since the area of the rectangle varies directly with its length and square of its width,

A ∝ lw 2

A = klw 2  ----(1)

Substitute A = 160, l = 5 and d = 4 to find the value k.

160 = k(5)(4 2 )

160 = k(5)(16 )

160 = 80k

Divide both sides by 80.

Substitute k = 2 in (1).

Substitute l  = 7, w = 3 and evaluate A.

A = 2(7)(3 2 )

A = 2(7)(9)

Area of the rectangle = 126 cm 2

Problem 5 :

The volume of a cylinder varies jointly as the square of radius and two times of its height. A cylinder with radius 4 cm and height 8 cm has a volume 128 π cm 3 . Find the volume of a cylinder with radius 3 cm and height 10 cm.

Let V represent volume of the cylinder, r represent radius and h represent height.

Since t he volume of a cylinder varies jointly as the radius and the sum of the radius and the height.

V ∝ r 2 (2h)

V = kr 2 (2h)   ----(1)

Substitute V = 128 π , r = 4 and h = 8 to find the value of k.

128π   = k(4 2 )(2 ⋅  8)

128π   = k(16)(16)

128π  = 256k

Divide both sides by 256.

π/2  = k

Substitute k = π/2 in (1).

V = ( π/2) r 2 (2h)

V = π r 2 h

Substitute r = 3, h = 10 and evaluate V.

V = π(3 2 )(10)

V = π(9) (10)

Volume of the cylinder = 90 π cm 3

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## How to Find Where the Graph of Parametric Equations Crosses Itself

Feb 21, 24 09:25 PM

## Average Rate of Change and Slope for Parametric Equations

Feb 20, 24 07:17 AM

## LCM Method for Time and Work

Feb 19, 24 07:32 PM

## Module 10: Rational and Radical Functions

Learning outcomes.

• Solve direct variation problems.
• Solve inverse variation problems.
• Solve problems involving joint variation.

A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance if she sells a vehicle for $4,600, she will earn$736. She wants to evaluate the offer, but she is not sure how. In this section we will look at relationships, such as this one, between earnings, sales, and commission rate.

## Direct Variation

In the example above, Nicole’s earnings can be found by multiplying her sales by her commission. The formula $e = 0.16s$ tells us her earnings, $e$, come from the product of 0.16, her commission, and the sale price of the vehicle, $s$. If we create a table, we observe that as the sales price increases, the earnings increase as well.

Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to$9,200, and we double the earnings from $736 to$1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called direct variation . Each variable in this type of relationship varies directly with the other.

The graph below represents the data for Nicole’s potential earnings. We say that earnings vary directly with the sales price of the car. The formula $y=k{x}^{n}$ is used for direct variation. The value $k$ is a nonzero constant greater than zero and is called the constant of variation . In this case, $k=0.16$ and $n=1$.

## A General Note: Direct Variation

If $x$   and $y$ are related by an equation of the form

$y=k{x}^{n}$

then we say that the relationship is direct variation and $y$  varies directly with the $n$th power of $x$. In direct variation relationships, there is a nonzero constant ratio $k=\dfrac{y}{{x}^{n}}$, where $k$ is called the constant of variation , which help defines the relationship between the variables.

## How To: Given a description of a direct variation problem, solve for an unknown.

• Identify the input, $x$, and the output, $y$.
• Determine the constant of variation. You may need to divide $y$ by the specified power of $x$ to determine the constant of variation.
• Use the constant of variation to write an equation for the relationship.
• Substitute known values into the equation to find the unknown.

## Example: Solving a Direct Variation Problem

The quantity $y$ varies directly with the cube of $x$. If $y=25$ when $x=2$, find $y$ when $x$ is 6.

The general formula for direct variation with a cube is $y=k{x}^{3}$. The constant can be found by dividing $y$ by the cube of $x$.

\begin{align} k&=\dfrac{y}{{x}^{3}} \\[1mm] &=\dfrac{25}{{2}^{3}}\\[1mm] &=\dfrac{25}{8}\end{align}

Now use the constant to write an equation that represents this relationship.

$y=\dfrac{25}{8}{x}^{3}$

Substitute $x=6$ and solve for $y$.

\begin{align}y&=\dfrac{25}{8}{\left(6\right)}^{3} \\[1mm] &=675\hfill \end{align}

## Analysis of the Solution

The graph of this equation is a simple cubic, as shown below.

Do the graphs of all direct variation equations look like Example 1?

No. Direct variation equations are power functions—they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through $(0, 0)$ .

The quantity $y$ varies directly with the square of $y$. If $y=24$ when $x=3$, find $y$ when $x$ is 4.

$\dfrac{128}{3}$

Watch this video to see a quick lesson in direct variation.  You will see more worked examples.

## Inverse and Joint Variation

Water temperature in an ocean varies inversely to the water’s depth. Between the depths of 250 feet and 500 feet, the formula $T=\Dfrac{14,000}{d}$ gives us the temperature in degrees Fahrenheit at a depth in feet below Earth’s surface. Consider the Atlantic Ocean, which covers 22% of Earth’s surface. At a certain location, at the depth of 500 feet, the temperature may be 28°F.

If we create a table we observe that, as the depth increases, the water temperature decreases.

We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be inversely proportional and each term varies inversely with the other. Inversely proportional relationships are also called inverse variations .

For our example, the graph depicts the inverse variation . We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula $y=\dfrac{k}{x}$ for inverse variation in this case uses $k=14,000$.

## A General Note: Inverse Variation

If $x$ and $y$ are related by an equation of the form

$y=\dfrac{k}{{x}^{n}}$

where $k$ is a nonzero constant, then we say that $y$  varies inversely with the $n$th power of $x$. In inversely proportional relationships, or inverse variations , there is a constant multiple $k={x}^{n}y$.

## Example: Writing a Formula for an Inversely Proportional Relationship

A tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.

Recall that multiplying speed by time gives distance. If we let $t$ represent the drive time in hours, and $v$ represent the velocity (speed or rate) at which the tourist drives, then $vt=$ distance. Because the distance is fixed at 100 miles, $vt=100$. Solving this relationship for the time gives us our function.

\begin{align}t\left(v\right)&=\dfrac{100}{v} \\[1mm] &=100{v}^{-1} \end{align}

We can see that the constant of variation is 100 and, although we can write the relationship using the negative exponent, it is more common to see it written as a fraction.

## How To: Given a description of an indirect variation problem, solve for an unknown.

• Determine the constant of variation. You may need to multiply $y$ by the specified power of $x$ to determine the constant of variation.

## Example: Solving an Inverse Variation Problem

A quantity $y$ varies inversely with the cube of $x$. If $y=25$ when $x=2$, find $y$ when $x$ is 6.

The general formula for inverse variation with a cube is $y=\dfrac{k}{{x}^{3}}$. The constant can be found by multiplying $y$ by the cube of $x$.

\begin{align}k&={x}^{3}y \\[1mm] &={2}^{3}\cdot 25 \\[1mm] &=200 \end{align}

Now we use the constant to write an equation that represents this relationship.

\begin{align}y&=\dfrac{k}{{x}^{3}},\hspace{2mm}k=200 \\[1mm] y&=\dfrac{200}{{x}^{3}} \end{align}

\begin{align}y&=\dfrac{200}{{6}^{3}} \\[1mm] &=\dfrac{25}{27} \end{align}

The graph of this equation is a rational function.

A quantity $y$ varies inversely with the square of $x$. If $y=8$ when $x=3$, find $y$ when $x$ is 4.

$\dfrac{9}{2}$

The following video presents a short lesson on inverse variation and includes more worked examples.

## Joint Variation

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation . For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable $c$, cost, varies jointly with the number of students, $n$, and the distance, $d$.

## A General Note: Joint Variation

Joint variation occurs when a variable varies directly or inversely with multiple variables.

For instance, if $x$ varies directly with both $y$ and $z$, we have $x=kyz$. If $x$ varies directly with $y$ and inversely with $z$, we have $x=\dfrac{ky}{z}$. Notice that we only use one constant in a joint variation equation.

## Example: Solving Problems Involving Joint Variation

A quantity $x$ varies directly with the square of $y$ and inversely with the cube root of $z$. If $x=6$ when $y=2$ and $z=8$, find $x$ when $y=1$ and $z=27$.

Begin by writing an equation to show the relationship between the variables.

$x=\dfrac{k{y}^{2}}{\sqrt[3]{z}}$

Substitute $x=6$, $y=2$, and $z=8$ to find the value of the constant $k$.

\begin{align}6&=\dfrac{k{2}^{2}}{\sqrt[3]{8}} \\[1mm] 6&=\dfrac{4k}{2} \\[1mm] 3&=k \end{align}

Now we can substitute the value of the constant into the equation for the relationship.

$x=\dfrac{3{y}^{2}}{\sqrt[3]{z}}$

To find $x$ when $y=1$ and $z=27$, we will substitute values for $y$ and $z$ into our equation.

\begin{align}x&=\dfrac{3{\left(1\right)}^{2}}{\sqrt[3]{27}} \\[1mm] &=1 \end{align}

$x$ varies directly with the square of $y$ and inversely with $z$. If $x=40$ when $y=4$ and $z=2$, find $x$ when $y=10$ and $z=25$.

$x=20$

The following video provides another worked example of a joint variation problem.

## Key Equations

Key concepts.

• A relationship where one quantity is a constant multiplied by another quantity is called direct variation.
• Two variables that are directly proportional to one another will have a constant ratio.
• A relationship where one quantity is a constant divided by another quantity is called inverse variation.
• Two variables that are inversely proportional to one another will have a constant multiple.
• In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation.
• Question ID 91391, 91393, 91394. Authored by : Michael Jenck. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
• Direct Variation Applications. Authored by : James Sousa (Mathispower4u.com) . Located at : https://youtu.be/plFOq4JaEyI . License : CC BY: Attribution
• Inverse Variation. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/awp2vxqd-l4 . License : CC BY: Attribution
• Joint Variation: Determine the Variation Constant (Volume of a Cone). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Provided by : Joint Variation: Determine the Variation Constant (Volume of a Cone). Located at : https://youtu.be/JREPATMScbM . License : CC BY: Attribution

## 4.8 Applications and Variation

Learning objectives.

• Solve applications involving uniform motion (distance problems).
• Solve work-rate applications.
• Set up and solve applications involving direct, inverse, and joint variation.

## Solving Uniform Motion Problems

Uniform motion (or distance) Described by the formula D = r t , where the distance D is given as the product of the average rate r and the time t traveled at that rate. problems involve the formula D = r t , where the distance D is given as the product of the average rate r and the time t traveled at that rate. If we divide both sides by the average rate r , then we obtain the formula

For this reason, when the unknown quantity is time, the algebraic setup for distance problems often results in a rational equation. We begin any uniform motion problem by first organizing our data with a chart. Use this information to set up an algebraic equation that models the application.

Sally traveled 15 miles on the bus and then another 72 miles on a train. The train was 18 miles per hour faster than the bus, and the total trip took 2 hours. What was the average speed of the train?

First, identify the unknown quantity and organize the data.

Let x represent the average speed (in miles per hour) of the bus.

Let x + 18 represent the average speed of the train.

To avoid introducing two more variables for the time column, use the formula t = D r . The time for each leg of the trip is calculated as follows:

T i m e   s p e n t   o n   t h e   b u s : t = D r = 15 x T i m e   s p e n t   o n   t h e   t r a i n : t = D r = 72 x + 18

Use these expressions to complete the chart.

The algebraic setup is defined by the time column. Add the time spent on each leg of the trip to obtain a total of 2 hours:

We begin solving this equation by first multiplying both sides by the LCD, x ( x + 18 ) .

15 x + 72 x + 18 = 2 x ( x + 18 ) ⋅ ( 15 x + 72 x + 18 ) = x ( x + 18 ) ⋅ 2 x ( x + 18 ) ⋅ 15 x + x ( x + 18 ) ⋅ 72 x + 18 = x ( x + 18 ) ⋅ 2 15 ( x + 18 ) + 72 x = 2 x ( x + 18 ) 15 x + 270 + 72 x = 2 x 2 + 36 x 87 x + 270 = 2 x 2 + 36 x 0 = 2 x 2 − 51 x − 270

Solve the resulting quadratic equation by factoring.

0 = 2 x 2 − 51 x − 270 0 = ( 2 x + 9 ) ( x − 30 ) 2 x + 9 =   0 or x − 30 = 0 x = − 9 2 x = 30

Since we are looking for an average speed we will disregard the negative answer and conclude the bus averaged 30 mph. Substitute x = 30 in the expression identified as the speed of the train.

x + 18 = 30 + 18 = 48

Answer: The speed of the train was 48 mph.

A boat can average 12 miles per hour in still water. On a trip downriver the boat was able to travel 29 miles with the current. On the return trip the boat was only able to travel 19 miles in the same amount of time against the current. What was the speed of the current?

First, identify the unknown quantities and organize the data.

Let c represent the speed of the river current.

Next, organize the given data in a chart. Traveling downstream, the current will increase the speed of the boat, so it adds to the average speed of the boat. Traveling upstream, the current slows the boat, so it will subtract from the average speed of the boat.

Use the formula t = D r to fill in the time column.

t r i p   d o w n r i v e r : t = D r = 29 12 + c t r i p   u p r i v e r : t = D r = 19 12 − c

Because the boat traveled the same amount of time downriver as it did upriver, finish the algebraic setup by setting the expressions that represent the times equal to each other.

29 12 + c = 19 12 − c

Since there is a single algebraic fraction on each side, we can solve this equation using cross multiplication.

29 12 + c = 19 12 − c 29 ( 12 − c ) = 19 ( 12 + c ) 348 − 29 c = 228 + 19 c 120 = 48 c 120 48 = c 5 2 = c

Answer: The speed of the current was 2 1 2 miles per hour.

Try this! A jet aircraft can average 160 miles per hour in calm air. On a trip, the aircraft traveled 600 miles with a tailwind and returned the 600 miles against a headwind of the same speed. If the total round trip took 8 hours, then what was the speed of the wind?

## Solving Work-Rate Problems

The rate at which a task can be performed is called a work rate The rate at which a task can be performed. . For example, if a painter can paint a room in 6 hours, then the task is to paint the room, and we can write

1     task 6   hours          work rate

In other words, the painter can complete 1 6 of the task per hour. If he works for less than 6 hours, then he will perform a fraction of the task. If he works for more than 6 hours, then he can complete more than one task. For example,

w o r k - r a t e       ×       t i m e       =   a m o u n t   o f   t a s k   c o m p l e t e d   1   6             ×           3     h r s           =         1 2   o n e - h a l f   o f   t h e   r o o m   p a i n t e d   1   6             ×           6     h r s           =         1   o n e   w h o l e   r o o m   p a i n t e d             1   6             ×             12     h r s     =         2   t w o   w h o l e   r o o m s   p a i n t e d

Obtain the amount of the task completed by multiplying the work rate by the amount of time the painter works. Typically, work-rate problems involve people or machines working together to complete tasks. In general, if t represents the time two people work together, then we have the following work-rate formula 1 t 1 ⋅ t + 1 t 2 ⋅ t = 1 , where 1 t 1 and 1 t 2 are the individual work rates and t is the time it takes to complete the task working together. :

1 t 1 t + 1 t 2 t =   a m o u n t   o f   t a s k   c o m p l e t e d   t o g e t h e r

Here 1 t 1 and 1 t 2 are the individual work rates.

Joe can paint a typical room in 2 hours less time than Mark. If Joe and Mark can paint 5 rooms working together in a 12 hour shift, how long does it take each to paint a single room?

Let x represent the time it takes Mark to paint a typical room.

Let x − 2 represent the time it takes Joe to paint a typical room.

Therefore, Mark’s individual work-rate is 1 x rooms per hour and Joe’s is 1 x − 2 rooms per hour. Both men worked for 12 hours. We can organize the data in a chart, just as we did with distance problems.

Working together, they can paint 5 total rooms in 12 hours. This leads us to the following algebraic setup:

12 x − 2 + 12 x = 5

Multiply both sides by the LCD, x ( x − 2 ) .

x ( x − 2 ) ⋅ ( 12 x − 2 + 12 x ) = x ( x − 2 ) ⋅ 5 x ( x − 2 ) ⋅ 12 x − 2 + x ( x − 2 ) ⋅ 12 x = x ( x − 2 ) ⋅ 5 12 x + 12 ( x − 2 ) = 5 x ( x − 2 ) 12 x + 12 x − 24 = 5 x 2 − 10 x 0 = 5 x 2 − 34 x + 24

0 = 5 x 2 − 34 x + 24 0 = ( 5 x − 4 ) ( x − 6 ) 5 x − 4 = 0 o r x − 6 = 0 5 x = 4 x = 6 x = 4 5

We can disregard 4 5 because back substituting into x − 2 would yield a negative time to paint a room. Take x = 6 to be the only solution and use it to find the time it takes Joe to paint a typical room.

x − 2 = 6 − 2 = 4

Answer: Joe can paint a typical room in 4 hours and Mark can paint a typical room in 6 hours. As a check we can multiply both work rates by 12 hours to see that together they can paint 5 rooms.

J o e 1     r o o m 4   h r s ⋅ 12   h r s = 3   r o o m s M a r k 1   r o o m 6   h r s ⋅ 12   h r s = 2   r o o m s   }   T o t a l   5   r o o m s    ✓

It takes Bill twice as long to lay a tile floor by himself as it does Manny. After working together with Bill for 4 hours, Manny was able to complete the job in 2 additional hours. How long would it have taken Manny working alone?

Let x represent the time it takes Manny to lay the floor alone.

Let 2 x represent the time it takes Bill to lay the floor alone.

Manny’s work rate is 1 x of the floor per hour and Bill’s work rate is 1 2 x . Bill worked on the job for 4 hours and Manny worked on the job for 6 hours.

This leads us to the following algebraic setup:

1 x ⋅ 6 + 1 2 x ⋅ 4 = 1

6 x + 4 2 x = 1 x ⋅ ( 6 x + 2 x ) = x ⋅ 1 6 + 2 = x 8 = x

Answer: It would have taken Manny 8 hours to complete the floor by himself.

Consider the work-rate formula where one task is to be completed.

1 t 1 t + 1 t 2 t = 1

Factor out the time t and then divide both sides by t . This will result in equivalent specialized work-rate formulas:

t ( 1 t 1 + 1 t 2 ) = 1 1 t 1 + 1 t 2 = 1 t

In summary, we have the following equivalent work-rate formulas:

Try this! Matt can tile a countertop in 2 hours, and his assistant can do the same job in 3 hours. If Matt starts the job and his assistant joins him 1 hour later, then how long will it take to tile the countertop?

## Solving Problems involving Direct, Inverse, and Joint variation

Many real-world problems encountered in the sciences involve two types of functional relationships. The first type can be explored using the fact that the distance s in feet an object falls from rest, without regard to air resistance, can be approximated using the following formula:

Here t represents the time in seconds the object has been falling. For example, after 2 seconds the object will have fallen s = 16 ( 2 ) 2 = 16 ⋅ 4 = 64 feet.

In this example, we can see that the distance varies over time as the product of a constant 16 and the square of the time t . This relationship is described as direct variation Describes two quantities x and y that are constant multiples of each other: y = k x . and 16 is called the constant of variation The nonzero multiple k , when quantities vary directly or inversely. . Furthermore, if we divide both sides of s = 16 t 2 by t 2 we have

In this form, it is reasonable to say that s is proportional to t 2 , and 16 is called the constant of proportionality Used when referring to the constant of variation. . In general, we have

Used when referring to direct variation.

Here k is nonzero and is called the constant of variation or the constant of proportionality. Typically, we will be given information from which we can determine this constant.

An object’s weight on Earth varies directly to its weight on the Moon. If a man weighs 180 pounds on Earth, then he will weigh 30 pounds on the Moon. Set up an algebraic equation that expresses the weight on Earth in terms of the weight on the Moon and use it to determine the weight of a woman on the Moon if she weighs 120 pounds on Earth.

Let y represent weight on Earth.

Let x represent weight on the Moon.

We are given that the “weight on Earth varies directly to the weight on the Moon.”

To find the constant of variation k , use the given information. A 180-lb man on Earth weighs 30 pounds on the Moon, or y   =   180 when   x   =   30 .

180 = k ⋅ 30

Solve for k .

180 30 = k 6 = k

Next, set up a formula that models the given information.

This implies that a person’s weight on Earth is 6 times his weight on the Moon. To answer the question, use the woman’s weight on Earth, y = 120 lbs, and solve for x .

120 = 6 x 120 6 = x 20 = x

Answer: The woman weighs 20 pounds on the Moon.

The second functional relationship can be explored using the formula that relates the intensity of light I to the distance from its source d .

Here k represents some constant. A foot-candle is a measurement of the intensity of light. One foot-candle is defined to be equal to the amount of illumination produced by a standard candle measured one foot away. For example, a 125-Watt fluorescent growing light is advertised to produce 525 foot-candles of illumination. This means that at a distance d = 1 foot, I = 525 foot-candles and we have:

525 = k ( 1 ) 2 525 = k

Using k = 525 we can construct a formula which gives the light intensity produced by the bulb:

I = 525 d 2

Here d represents the distance the growing light is from the plants. In the following chart, we can see that the amount of illumination fades quickly as the distance from the plants increases.

This type of relationship is described as an inverse variation Describes two quantities x and y , where one variable is directly proportional to the reciprocal of the other: y = k x . . We say that I is inversely proportional Used when referring to inverse variation. to the square of the distance d , where 525 is the constant of proportionality. In general, we have

Again, k is nonzero and is called the constant of variation or the constant of proportionality.

The weight of an object varies inversely as the square of its distance from the center of Earth. If an object weighs 100 pounds on the surface of Earth (approximately 4,000 miles from the center), how much will it weigh at 1,000 miles above Earth’s surface?

Let w represent the weight of the object.

Let d represent the object’s distance from the center of Earth.

Since “ w varies inversely as the square of d ,” we can write

Use the given information to find k . An object weighs 100 pounds on the surface of Earth, approximately 4,000 miles from the center. In other words, w = 100 when d = 4,000:

100 = k ( 4,000 ) 2

( 4,000 ) 2 ⋅ 100 = ( 4,000 ) 2 ⋅ k ( 4,000 ) 2 1,600,000,000 = k 1.6 × 10 9 = k

Therefore, we can model the problem with the following formula:

w = 1.6 × 10 9 d 2

To use the formula to find the weight, we need the distance from the center of Earth. Since the object is 1,000 miles above the surface, find the distance from the center of Earth by adding 4,000 miles:

d = 4,000 + 1,000 = 5,000     miles

To answer the question, use the formula with d = 5,000.

y = 1.6 × 10 9 ( 5,000 ) 2 = 1.6 × 10 9 25,000,000 = 1.6 × 10 9 2.5 × 10 7 = 0.64 × 10 2 = 64

Answer: The object will weigh 64 pounds at a distance 1,000 miles above the surface of Earth.

Lastly, we define relationships between multiple variables, described as joint variation Describes a quantity y that varies directly as the product of two other quantities x and z : y = k x z . . In general, we have

Used when referring to joint variation.

Here k is nonzero and is called the constant of variation or the constant of proportionality.

The area of an ellipse varies jointly as a , half of the ellipse’s major axis, and b , half of the ellipse’s minor axis as pictured. If the area of an ellipse is 300 π   cm 2 , where a = 10   cm and b = 30   cm , what is the constant of proportionality? Give a formula for the area of an ellipse.

If we let A represent the area of an ellipse, then we can use the statement “area varies jointly as a and b ” to write

To find the constant of variation k , use the fact that the area is 300 π when a = 10 and b = 30 .

300 π = k ( 10 ) ( 30 ) 300 π = 300 k π = k

Therefore, the formula for the area of an ellipse is

Answer: The constant of proportionality is π and the formula for the area of an ellipse is A = a b π .

Try this! Given that y varies directly as the square of x and inversely with z , where y = 2 when x = 3 and z = 27, find y when x = 2 and z = 16.

## Key Takeaways

• When solving distance problems where the time element is unknown, use the equivalent form of the uniform motion formula, t = D r , to avoid introducing more variables.
• When solving work-rate problems, multiply the individual work rate by the time to obtain the portion of the task completed. The sum of the portions of the task results in the total amount of work completed.
• The setup of variation problems usually requires multiple steps. First, identify the key words to set up an equation and then use the given information to find the constant of variation k . After determining the constant of variation, write a formula that models the problem. Once a formula is found, use it to answer the question.

## Topic Exercises

Part a: solving uniform motion problems.

Use algebra to solve the following applications.

Every morning Jim spends 1 hour exercising. He runs 2 miles and then he bikes 16 miles. If Jim can bike twice as fast as he can run, at what speed does he average on his bike?

Sally runs 3 times as fast as she walks. She ran for 3 4 of a mile and then walked another 3 1 2 miles. The total workout took 1 1 2 hours. What was Sally’s average walking speed?

On a business trip, an executive traveled 720 miles by jet and then another 80 miles by helicopter. If the jet averaged 3 times the speed of the helicopter, and the total trip took 4 hours, what was the average speed of the jet?

A triathlete can run 3 times as fast as she can swim and bike 6 times as fast as she can swim. The race consists of a 1 4 mile swim, 3 mile run, and a 12 mile bike race. If she can complete all of these events in 1 5 8 hour, then how fast can she swim, run and bike?

On a road trip, Marty was able to drive an average 4 miles per hour faster than George. If Marty was able to drive 39 miles in the same amount of time George drove 36 miles, what was Marty’s average speed?

The bus is 8 miles per hour faster than the trolley. If the bus travels 9 miles in the same amount of time the trolley can travel 7 miles, what is the average speed of each?

Terry decided to jog the 5 miles to town. On the return trip, she walked the 5 miles home at half of the speed that she was able to jog. If the total trip took 3 hours, what was her average jogging speed?

James drove the 24 miles to town and back in 1 hour. On the return trip, he was able to average 20 miles per hour faster than he averaged on the trip to town. What was his average speed on the trip to town?

A light aircraft was able to travel 189 miles with a 14 mile per hour tailwind in the same time it was able to travel 147 miles against it. What was the speed of the aircraft in calm air?

A jet flew 875 miles with a 30 mile per hour tailwind. On the return trip, against a 30 mile per hour headwind, it was able to cover only 725 miles in the same amount of time. How fast was the jet in calm air?

A helicopter averaged 90 miles per hour in calm air. Flying with the wind it was able to travel 250 miles in the same amount of time it took to travel 200 miles against it. What is the speed of the wind?

Mary and Joe took a road-trip on separate motorcycles. Mary’s average speed was 12 miles per hour less than Joe’s average speed. If Mary drove 115 miles in the same time it took Joe to drive 145 miles, what was Mary’s average speed?

A boat averaged 12 miles per hour in still water. On a trip downstream, with the current, the boat was able to travel 26 miles. The boat then turned around and returned upstream 33 miles. How fast was the current if the total trip took 5 hours?

If the river current flows at an average 3 miles per hour, a tour boat can make an 18-mile tour downstream with the current and back the 18 miles against the current in 4 1 2 hours. What is the average speed of the boat in still water?

Jose drove 10 miles to his grandmother’s house for dinner and back that same evening. Because of traffic, he averaged 20 miles per hour less on the return trip. If it took 1 4 hour longer to get home, what was his average speed driving to his grandmother’s house?

Jerry paddled his kayak, upstream against a 1 mph current, for 12 miles. The return trip, downstream with the 1 mph current, took one hour less time. How fast did Jerry paddle the kayak in still water?

James and Mildred left the same location in separate cars and met in Los Angeles 300 miles away. James was able to average 10 miles an hour faster than Mildred on the trip. If James arrived 1 hour earlier than Mildred, what was Mildred’s average speed?

A bus is 20 miles per hour faster than a bicycle. If Bill boards a bus at the same time and place that Mary departs on her bicycle, Bill will arrive downtown 5 miles away 1 3 hour earlier than Mary. What is the average speed of the bus?

## Part B: Solving Work-Rate Problems

Mike can paint the office by himself in 4 1 2 hours. Jordan can paint the office in 6 hours. How long will it take them to paint the office working together?

Barry can lay a brick driveway by himself in 3 1 2 days. Robert does the same job in 5 days. How long will it take them to lay the brick driveway working together?

A larger pipe fills a water tank twice as fast as a smaller pipe. When both pipes are used, they fill the tank in 10 hours. If the larger pipe is left off, how long would it take the smaller pipe to fill the tank?

A newer printer can print twice as fast as an older printer. If both printers working together can print a batch of flyers in 45 minutes, then how long would it take the older printer to print the batch working alone?

Mary can assemble a bicycle for display in 2 hours. It takes Jane 3 hours to assemble a bicycle. How long will it take Mary and Jane, working together, to assemble 5 bicycles?

Working alone, James takes twice as long to assemble a computer as it takes Bill. In one 8-hour shift, working together, James and Bill can assemble 6 computers. How long would it take James to assemble a computer if he were working alone?

Working alone, it takes Harry one hour longer than Mike to install a fountain. Together they can install 10 fountains in 12 hours. How long would it take Mike to install 10 fountains by himself?

Working alone, it takes Henry 2 hours longer than Bill to paint a room. Working together they painted 2 1 2 rooms in 6 hours. How long would it have taken Henry to paint the same amount if he were working alone?

Manny, working alone, can install a custom cabinet in 3 hours less time than his assistant. Working together they can install the cabinet in 2 hours. How long would it take Manny to install the cabinet working alone?

Working alone, Garret can assemble a garden shed in 5 hours less time than his brother. Working together, they need 6 hours to build the garden shed. How long would it take Garret to build the shed working alone?

Working alone, the assistant-manager takes 2 more hours than the manager to record the inventory of the entire shop. After working together for 2 hours, it took the assistant-manager 1 additional hour to complete the inventory. How long would it have taken the manager to complete the inventory working alone?

An older printer can print a batch of sales brochures in 16 minutes. A newer printer can print the same batch in 10 minutes. After working together for some time, the newer printer was shut down and it took the older printer 3 more minutes to complete the job. How long was the newer printer operating?

## Part C: Solving Variation Problems

Translate each of the following sentences into a mathematical formula.

The distance D an automobile can travel is directly proportional to the time t that it travels at a constant speed.

The extension of a hanging spring d is directly proportional to the weight w attached to it.

An automobile’s braking distance d is directly proportional to the square of the automobile’s speed v .

The volume V of a sphere varies directly as the cube of its radius r .

The volume V of a given mass of gas is inversely proportional to the pressure p exerted on it.

Every particle of matter in the universe attracts every other particle with a force F that is directly proportional to the product of the masses m 1 and m 2 of the particles, and it is inversely proportional to the square of the distance d between them.

Simple interest I is jointly proportional to the annual interest rate r and the time t in years a fixed amount of money is invested.

The time t it takes an object to fall is directly proportional to the square root of the distance d it falls.

Construct a mathematical model given the following:

y varies directly as x , and y = 30 when x = 6.

y varies directly as x , and y = 52 when x = 4.

y is directly proportional to x , and y = 12 when x = 3.

y is directly proportional to x , and y = 120 when x = 20.

y is inversely proportional to x , and y = 3 when x = 9.

y is inversely proportional to x , and y = 21 when x = 3.

y varies inversely as x , and y = 2 when x = 1 8 .

y varies inversely as x , and y = 3 2 when x = 1 9 .

y is jointly proportional to x and z , where y = 2 when x = 1 and z = 3.

y is jointly proportional to x and z , where y = 15 when x = 3 and z = 7.

y varies jointly as x and z , where y = 2 3 when x = 1 2 and z = 12.

y varies jointly as x and z , where y = 5 when x = 3 2 and z = 2 9 .

y varies directly as the square of x , where y = 45 when x = 3.

y varies directly as the square of x , where y = 3 when x = 1 2 .

y is inversely proportional to the square of x , where y = 27 when x = 1 3 .

y is inversely proportional to the square of x , where y = 9 when x = 2 3 .

y varies jointly as x and the square of z , where y = 6 when x = 1 4 and z = 2 3 .

y varies jointly as x and z and inversely as the square of w , where y = 5 when x = 1, z = 3, and w = 1 2 .

y varies directly as the square root of x and inversely as the square of z , where y = 15 when x = 25 and z = 2.

y varies directly as the square of x and inversely as z and the square of w , where y = 14 when x = 4, w = 2, and z = 2.

Solve applications involving variation.

Revenue in dollars is directly proportional to the number of branded sweatshirts sold. The revenue earned from selling 25 sweatshirts is $318.75. Determine the revenue if 30 sweatshirts are sold. The sales tax on the purchase of a new car varies directly as the price of the car. If an$18,000 new car is purchased, then the sales tax is $1,350. How much sales tax is charged if the new car is priced at$22,000?

The price of a share of common stock in a company is directly proportional to the earnings per share (EPS) of the previous 12 months. If the price of a share of common stock in a company is $22.55, and the EPS is published to be$1.10, determine the value of the stock if the EPS increases by $0.20. The distance traveled on a road trip varies directly with the time spent on the road. If a 126-mile trip can be made in 3 hours, then what distance can be traveled in 4 hours? The circumference of a circle is directly proportional to its radius. The circumference of a circle with radius 7 centimeters is measured as 14 π centimeters. What is the constant of proportionality? The area of circle varies directly as the square of its radius. The area of a circle with radius 7 centimeters is determined to be 49 π square centimeters. What is the constant of proportionality? The surface area of a sphere varies directly as the square of its radius. When the radius of a sphere measures 2 meters, the surface area measures 16 π square meters. Find the surface area of a sphere with radius 3 meters. The volume of a sphere varies directly as the cube of its radius. When the radius of a sphere measures 3 meters, the volume is 36 π cubic meters. Find the volume of a sphere with radius 1 meter. With a fixed height, the volume of a cone is directly proportional to the square of the radius at the base. When the radius at the base measures 10 centimeters, the volume is 200 cubic centimeters. Determine the volume of the cone if the radius of the base is halved. The distance d an object in free fall drops varies directly with the square of the time t that it has been falling. If an object in free fall drops 36 feet in 1.5 seconds, then how far will it have fallen in 3 seconds? Hooke’s law suggests that the extension of a hanging spring is directly proportional to the weight attached to it. The constant of variation is called the spring constant. Robert Hooke (1635—1703) A hanging spring is stretched 5 inches when a 20-pound weight is attached to it. Determine its spring constant. A hanging spring is stretched 3 centimeters when a 2-kilogram weight is attached to it. Determine the spring constant. If a hanging spring is stretched 3 inches when a 2-pound weight is attached, how far will it stretch with a 5-pound weight attached? If a hanging spring is stretched 6 centimeters when a 4-kilogram weight is attached to it, how far will it stretch with a 2-kilogram weight attached? The braking distance of an automobile is directly proportional to the square of its speed. It takes 36 feet to stop a particular automobile moving at a speed of 30 miles per hour. How much breaking distance is required if the speed is 35 miles per hour? After an accident, it was determined that it took a driver 80 feet to stop his car. In an experiment under similar conditions, it takes 45 feet to stop the car moving at a speed of 30 miles per hour. Estimate how fast the driver was moving before the accident. Robert Boyle (1627—1691) Boyle’s law states that if the temperature remains constant, the volume V of a given mass of gas is inversely proportional to the pressure p exerted on it. A balloon is filled to a volume of 216 cubic inches on a diving boat under 1 atmosphere of pressure. If the balloon is taken underwater approximately 33 feet, where the pressure measures 2 atmospheres, then what is the volume of the balloon? A balloon is filled to 216 cubic inches under a pressure of 3 atmospheres at a depth of 66 feet. What would the volume be at the surface, where the pressure is 1 atmosphere? To balance a seesaw, the distance from the fulcrum that a person must sit is inversely proportional to his weight. If a 72-pound boy is sitting 3 feet from the fulcrum, how far from the fulcrum must a 54-pound boy sit to balance the seesaw? The current I in an electrical conductor is inversely proportional to its resistance R . If the current is 1 4 ampere when the resistance is 100 ohms, what is the current when the resistance is 150 ohms? The amount of illumination I is inversely proportional to the square of the distance d from a light source. If 70 foot-candles of illumination is measured 2 feet away from a lamp, what level of illumination might we expect 1 2 foot away from the lamp? The amount of illumination I is inversely proportional to the square of the distance d from a light source. If 40 foot-candles of illumination is measured 3 feet away from a lamp, at what distance can we expect 10 foot-candles of illumination? The number of men, represented by y , needed to lay a cobblestone driveway is directly proportional to the area A of the driveway and inversely proportional to the amount of time t allowed to complete the job. Typically, 3 men can lay 1,200 square feet of cobblestone in 4 hours. How many men will be required to lay 2,400 square feet of cobblestone in 6 hours? The volume of a right circular cylinder varies jointly as the square of its radius and its height. A right circular cylinder with a 3-centimeter radius and a height of 4 centimeters has a volume of 36 π cubic centimeters. Find a formula for the volume of a right circular cylinder in terms of its radius and height. The period T of a pendulum is directly proportional to the square root of its length L . If the length of a pendulum is 1 meter, then the period is approximately 2 seconds. Approximate the period of a pendulum that is 0.5 meter in length. The time t it takes an object to fall is directly proportional to the square root of the distance d it falls. An object dropped from 4 feet will take 1 2 second to hit the ground. How long will it take an object dropped from 16 feet to hit the ground? Newton’s universal law of gravitation states that every particle of matter in the universe attracts every other particle with a force F that is directly proportional to the product of the masses m 1 and m 2 of the particles and inversely proportional to the square of the distance d between them. The constant of proportionality is called the gravitational constant. Sir Isaac Newton (1643—1727) Source: Portrait of Isaac Newton by Sir Godfrey Kneller, from http://commons.wikimedia.org/wiki/File:GodfreyKneller-IsaacNewton-1689. http://commons.wikimedia.org/wiki/File:Frans_Hals_-_Portret_ _van_Ren%C3%A9_Descartes.jpg. If two objects with masses 50 kilograms and 100 kilograms are 1 2 meter apart, then they produce approximately 1.34 × 10 − 6 newtons (N) of force. Calculate the gravitational constant. Use the gravitational constant from the previous exercise to write a formula that approximates the force F in newtons between two masses m 1 and m 2 , expressed in kilograms, given the distance d between them in meters. Calculate the force in newtons between Earth and the Moon, given that the mass of the Moon is approximately 7.3 × 10 22 kilograms, the mass of Earth is approximately 6.0 × 10 24 kilograms, and the distance between them is on average 1.5 × 10 11 meters. Calculate the force in newtons between Earth and the Sun, given that the mass of the Sun is approximately 2.0 × 10 30 kilograms, the mass of Earth is approximately 6.0 × 10 24 kilograms, and the distance between them is on average 3.85 × 10 8 meters. If y varies directly as the square of x , then how does y change if x is doubled? If y varies inversely as square of t , then how does y change if t is doubled? If y varies directly as the square of x and inversely as the square of t , then how does y change if both x and t are doubled? 20 miles per hour 240 miles per hour 52 miles per hour 5 miles per hour 112 miles per hour 10 miles per hour 1 mile per hour 40 miles per hour 50 miles per hour 2 4 7 hours y = 2 3 x z y = 1 9 x z • y = 54 x z 2 • y = 12 x z 2 36 π square meters 50 cubic centimeters 108 cubic inches 1,120 foot-candles 1.4 seconds 6.7 × 10 − 11 N m 2 / kg 2 1.98 × 10 20 N y changes by a factor of 4 y remains unchanged ## Joint Variation ## What is Joint Variation? Joint variation occurs when a variable changes in relation to two or more other variables. In mathematics, joint variation is a situation where a quantity varies directly as the product of two or more other quantities. For example, if a variable z varies directly as x and y , then we can say that z varies jointly with x and y . ## Formula for Joint Variation The general formula for joint variation is: • z is the variable that jointly varies, • x and y are the variables that z varies jointly with, • k is the constant of proportionality. In joint variation, the value of z is directly proportional to the values of x and y taken together. This means that if x or y (or both) increase, z will also increase, assuming that k remains constant. Similarly, if x or y decrease, z will decrease as well. ## Examples of Joint Variation Joint variation can be seen in many real-world scenarios. Here are a few examples: • The volume V of a gas varies jointly with the temperature T and the pressure P , as described by the ideal gas law PV = nRT , where n is the number of moles of gas and R is the ideal gas constant. • In physics, the force F exerted by an object varies jointly with its mass m and acceleration a , as described by Newton's second law F = ma . • The electrical resistance R of a wire varies jointly with the resistivity ρ of the material and the length L of the wire, inversely with the cross-sectional area A , as described by the formula R = ρL/A . ## Solving Joint Variation Problems To solve joint variation problems, one must first determine the constant of proportionality k . This is usually done by using a set of known values for x , y , and z . Once k is known, the formula can be used to find unknown values given the other known quantities. For example, if a problem states that z varies jointly with x and y , and z equals 12 when x is 2 and y is 3, we can find k by substituting these values into the formula: 12 = k(2)(3) Solving for k , we get: Now, with the constant k known, we can predict z for any other values of x and y . Understanding joint variation is essential for solving problems in various fields of science, engineering, and mathematics. It allows us to describe how quantities change together and to predict the behavior of one variable based on the behavior of others. By mastering joint variation, one can gain deeper insights into the relationships between different variables in complex systems. 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Free from preassure and study anxiety. ## Joint, Inverse, and Combined Variation Abilities covered in this lesson, lesson settings, catch-up and review. Here are a few recommended readings before getting started with this lesson. • Reciprocal Functions • Graphing Reciprocal Functions • Direct Variation and Proportional Relationships ## Number of Phone Calls Per Day Between Two Cities The average number of phone calls per day between two cities N varies directly with the populations of the cities P_1 and P_2, and inversely with the square of distance d between the two cities. ## Joint Variation A joint variation , also known as joint proportionality , occurs when one variable varies directly with two or more variables . In other words, if a variable varies directly with the product of other variables, it is called joint variation. Here, the variable z varies jointly with x and y, and k is the constant of variation . Here are some examples of joint variation. ## Television Series Vincenzo and Emily are having a lively chat about television series they love. Emily managed to watch 164 episodes of The Flash in just 50 days! Each episode typically lasts 40 minutes. Use the fact that if z varies jointly with x and y, the equation of variation is z=k xy, where k is the constant of variation . Substitute values .LHS /6560.=.RHS /6560. a/b=.a /10./.b /10. Rearrange equation a/c* b = a* b/c Calculate quotient Round to nearest integer ## Recognzing Inverse Variation Inverse variation. An inverse variation , or inverse proportionality , occurs when two non-zero variables have a relationship such that their product is constant . This relationship is often written with one of the variables isolated on the left-hand side. xy=k or y=k/x ## Identifying the Type of Variation Determine whether the relationship between the variables in the table shows direct or inverse variation, or neither. ## Number of Songs on Emily's Phone Emily, tired of watching shows, wants to update the playlist on her phone before starting a family road trip from Portland to San Francisco. The number of songs that can be stored on her phone varies inversely with the average size of a song. Emily's phone can store 4100 songs when the average size of a song is 4 megabytes (MB). x= 4, y= 4100 LHS * 4=RHS* 4 y=16 400/x Now a table that shows the number of songs when the average size of a song is 3 MB, 4 MB, 5 MB, and 6 MB can be made. In the table, as the size gets larger, the number of songs that the phone can store gets smaller. Therefore, the number of songs decreases as the average size increases. ## Emily's Trip to San Francisco Example Graph: t=k/r Here, k is a constant of variation. It is also given that Emily lives 450 miles away from San Francisco. This is the value of k. Recall that a distance is a product of the time and rate of speed . Thus, the time can be expressed as a quotient of the distance by the rate of speed. d=rt ⇒ t=d/r Comparing this formula with the equation, it is seen that k indeed represents the distance. Therefore, k can be substituted with 640. t=640/r Now make a table of values to graph the equation. Ordered pairs (r,t) are the coordinates of the points on the graph. Plot the points and connect them with a smooth curve. LHS * r=RHS* r .LHS /12.=.RHS /12. ## Combined Variation A combined variation , or combined proportionality , occurs when one variable depends on two or more variables, either directly , inversely , or a combination of both. This means that any joint variation is also a combined variation. The variable z varies directly with x and inversely with y, and k is the constant of variation . Therefore, this is a combined variation. Here are some examples. ## Number of T-Shirts Sold Emily is wandering around a gift shop to buy gifts for some of her friends. Emily overhears a conversation between the shopkeeper and an employee. The shopkeeper says that the number of t-shirts sold is directly proportional to their advertising budget and inversely proportional to the price of each t-shirt. When$1200 are spent on advertising and the price of each t-shirt is $4.80, the number of t-shirts sold is 6500. How many t-shirts are sold when the advertising budget is$1800 and the price of each t-shirt is $6? Use the equation of the combined variation , z = kxy, where k is the constant of variation . When one quantity varies with respect to two or more quantities, this variation can be regarded as a combined variation. LHS * 4.80=RHS* 4.80 .LHS /1200.=.RHS /1200. b= 1800, p= 6 ## Alternative Solution Cross multiply .LHS /7200.=.RHS /7200. ## Finding the Value of z In the applet, various types of variations are shown randomly. Find the value of z by using the given values. If necessary, round the answer to the two decimal places. In this lesson, variation types are explained with real-life examples. Considering those examples, the challenge presented at the beginning of the lesson can be solved with confidence. Recall that the average number of phone calls per day between two cities varies directly with the populations of the cities and inversely with the square of the distance between the two cities. N: & Number of phone calls per day P_1: & Population of one of the cities P_2: & Population of the other city d : & Distance between the cities It is known that N varies directly with P_1 and P_2, and inversely with the square of d. Therefore, N is equal to the product of k, P_1, and P_2 divided by the square of d. N=k P_1 P_2/d^2 Here k is the constant of variation and cannot be 0. This equation models the given variation. Calculate power a* b/c=a*b/c .LHS /1 116 000.=.RHS /1 116 000. Round to 3 decimal place(s) N= 0.038 P_1 P_2/d^2 For clarity, make a table to organize the given information. LHS * d^2=RHS* d^2 .LHS /806 000.=.RHS /806 000. sqrt(LHS)=sqrt(RHS) sqrt(a^2)=± a Calculate root ## Recommended exercises • school Campus Bookshelves • menu_book Bookshelves • perm_media Learning Objects • login Login • how_to_reg Request Instructor Account • hub Instructor Commons • Download Page (PDF) • Download Full Book (PDF) • Periodic Table • Physics Constants • Scientific Calculator • Reference & Cite • Tools expand_more • Readability selected template will load here This action is not available. ## 2.10: Modeling Using Variation • Last updated • Save as PDF • Page ID 134149 Learning Objectives • Solve direct variation problems. • Solve inverse variation problems. • Solve problems involving joint variation. A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for$4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate. ## Solving Direct Variation Problems In the example above, Nicole’s earnings can be found by multiplying her sales by her commission. The formula $$e=0.16s$$ tells us her earnings, $$e$$, come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive. See Table 5.8.1 . Table 5.8.1 Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from$4,600 to $9,200, and we double the earnings from$736 to 1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called direct variation . Each variable in this type of relationship varies directly with the other. Figure 5.8.1 represents the data for Nicole’s potential earnings. We say that earnings vary directly with the sales price of the car. The formula $$y=kx^n$$ is used for direct variation. The value $$k$$ is a nonzero constant greater than zero and is called the constant of variation . In this case, $$k=0.16$$ and $$n=1$$. We saw functions like this one when we discussed power functions. Figure has invalid source: image visible until saved.... src="/@api/deki/pages/=Bookshelves%252FPrecalculus%252FBook%25253A_Precalculus_(OpenStax)%252F03%25253A_Polynomial_and_Rational_Functions%252F3.9%25253A_Modeling_Using_Variation/files/CNX_Precalc_Figure_03_09_001.jpg A General Note: DIRECT VARIATION If $$x$$ and $$y$$ are related by an equation of the form then we say that the relationship is direct variation and $$y$$ varies directly with, or is proportional to, the $$n$$th power of $$x$$. In direct variation relationships, there is a nonzero constant ratio $$k=\dfrac{y}{x^n}$$, where $$k$$ is called the constant of variation , which help defines the relationship between the variables. Given a description of a direct variation problem, solve for an unknown. • Identify the input, $$x$$,and the output, $$y$$. • Determine the constant of variation. You may need to divide $$y$$ by the specified power of $$x$$ to determine the constant of variation. • Use the constant of variation to write an equation for the relationship. • Substitute known values into the equation to find the unknown. Solving a Direct Variation Problem The quantity $$y$$ varies directly with the cube of $$x$$. If $$y=25$$ when $$x=2$$, find $$y$$ when $$x$$ is $$6$$. The general formula for direct variation with a cube is $$y=kx^3$$. The constant can be found by dividing $$y$$ by the cube of $$x$$. $$k=\dfrac{y}{x^3}$$ $$=\dfrac{25}{2^3}$$ $$=\dfrac{25}{8}$$ Now use the constant to write an equation that represents this relationship. $$y=\dfrac{25}{8}x^3$$ Substitute $$x=6$$ and solve for $$y$$. $$y=\dfrac{25}{8}{(6)}^3$$ The graph of this equation is a simple cubic, as shown in Figure 5.8.2 . Do the graphs of all direct variation equations look like Example ? No. Direct variation equations are power functions—they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through $$(0,0)$$. The quantity $$y$$ varies directly with the square of $$x$$. If $$y=24$$ when $$x=3$$, find $$y$$ when $$x$$ is 4. $$\frac{128}{3}$$ ## Solving Inverse Variation Problems Water temperature in an ocean varies inversely to the water’s depth. The formula $$T=\frac{14,000}{d}$$ gives us the temperature in degrees Fahrenheit at a depth in feet below Earth’s surface. Consider the Atlantic Ocean, which covers 22% of Earth’s surface. At a certain location, at the depth of 500 feet, the temperature may be 28°F. If we create Table 5.8.2 , we observe that, as the depth increases, the water temperature decreases. Table 5.8.2 We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be inversely proportional and each term varies inversely with the other. Inversely proportional relationships are also called inverse variations . For our example, Figure 5.8.3 depicts the inverse variation. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula $$y=\frac{k}{x}$$ for inverse variation in this case uses $$k=14,000$$. A General Note: INVERSE VARIATION $$y=\frac{k}{x^n}$$ where $$k$$ is a nonzero constant, then we say that $$y$$ varies inversely with the $$n$$th power of $$x$$. In inversely proportional relationships, or inverse variations , there is a constant multiple $$k=x^ny$$. Writing a Formula for an Inversely Proportional Relationship A tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives. Recall that multiplying speed by time gives distance. If we let $$t$$ represent the drive time in hours, and $$v$$ represent the velocity (speed or rate) at which the tourist drives, then $$vt=$$distance. Because the distance is fixed at 100 miles, $$vt=100$$ so $$t=\frac{100}{v}$$. Because time is a function of velocity, we can write $$t(v)$$. $$t(v)=\frac{100}{v}$$ $$=100v^{−1}$$ We can see that the constant of variation is 100 and, although we can write the relationship using the negative exponent, it is more common to see it written as a fraction. We say that time varies inversely with velocity. Given a description of an indirect variation problem, solve for an unknown. • Identify the input, $$x$$, and the output, $$y$$. • Determine the constant of variation. You may need to multiply $$y$$ by the specified power of $$x$$ to determine the constant of variation. Solving an Inverse Variation Problem A quantity $$y$$ varies inversely with the cube of $$x$$. If $$y=25$$ when $$x=2$$, find $$y$$ when $$x$$ is $$6$$. The general formula for inverse variation with a cube is $$y=\frac{k}{x^3}$$. The constant can be found by multiplying $$y$$ by the cube of $$x$$. $$=2^3⋅25$$ Now we use the constant to write an equation that represents this relationship. $$y=\dfrac{k}{x^3}$$, $$k=200$$ $$y=\dfrac{200}{x^3}$$ $$y=\dfrac{200}{6^3}$$ $$=\dfrac{25}{27}$$ The graph of this equation is a rational function, as shown in Figure 5.8.4 . A quantity $$y$$ varies inversely with the square of $$x$$. If $$y=8$$ when $$x=3$$, find $$y$$ when $$x$$ is $$4$$. $$\frac{9}{2}$$ Solving Problems Involving Joint Variation Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation . For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable $$c$$,cost, varies jointly with the number of students, $$n$$,and the distance, $$d$$. A General Note: JOINT VARIATION Joint variation occurs when a variable varies directly or inversely with multiple variables. For instance, if $$x$$ varies directly with both $$y$$ and $$z$$, we have $$x=kyz$$. If $$x$$ varies directly with $$y$$ and inversely with $$z$$,we have $$x=\frac{ky}{z}$$. Notice that we only use one constant in a joint variation equation. A quantity $$x$$ varies directly with the square of $$y$$ and inversely with the cube root of $$z$$. If $$x=6$$ when $$y=2$$ and $$z=8$$, find $$x$$ when $$y=1$$ and $$z=27$$. Begin by writing an equation to show the relationship between the variables. $$x=\dfrac{ky^2}{\sqrt[3]{z}}$$ Substitute $$x=6$$, $$y=2$$, and $$z=8$$ to find the value of the constant $$k$$. $$6=\dfrac{k2^2}{\sqrt[3]{8}}$$ $$6=\dfrac{4k}{2}$$ Now we can substitute the value of the constant into the equation for the relationship. $$x=\dfrac{3y^2}{\sqrt[3]{z}}$$ To find $$x$$ when $$y=1$$ and $$z=27$$, we will substitute values for $$y$$ and $$z$$ into our equation. $$x=\dfrac{3{(1)}^2}{\sqrt[3]{27}}$$ A quantity $$x$$ varies directly with the square of $$y$$ and inversely with $$z$$. If $$x=40$$ when $$y=4$$ and $$z=2$$, find $$x$$ when $$y=10$$ and $$z=25$$. Access these online resources for additional instruction and practice with direct and inverse variation. • Direct Variation • Inverse Variation • Direct and Inverse Variation Visit this website for additional practice questions from Learningpod. ## Study Guides > College Algebra CoRequisite Course Inverse and joint variation, learning outcomes. • Solve an Inverse variation problem. • Write a formula for an inversely proportional relationship. ## A General Note: Inverse Variation Isolating the constant of variation, example: writing a formula for an inversely proportional relationship. \begin{align}t\left(v\right)&=\dfrac{100}{v} \\[1mm] &=100{v}^{-1} \end{align} ## How To: Given a description of an inverse variation problem, solve for an unknown. • Identify the input, $x$, and the output, $y$. • Determine the constant of variation. You may need to multiply $y$ by the specified power of $x$ to determine the constant of variation. • Use the constant of variation to write an equation for the relationship. • Substitute known values into the equation to find the unknown. ## Example: Solving an Inverse Variation Problem \begin{align}k&={x}^{3}y \\[1mm] &={2}^{3}\cdot 25 \\[1mm] &=200 \end{align} \begin{align}y&=\dfrac{k}{{x}^{3}},\hspace{2mm}k=200 \\[1mm] y&=\dfrac{200}{{x}^{3}} \end{align} \begin{align}y&=\dfrac{200}{{6}^{3}} \\[1mm] &=\dfrac{25}{27} \end{align} ## Analysis of the Solution Answer: [latex-display]\dfrac{9}{2}[/latex-display] ## Joint Variation A general note: joint variation, example: solving problems involving joint variation. $x=\dfrac{k{y}^{2}}{\sqrt[3]{z}}$ \begin{align}6&=\dfrac{k{2}^{2}}{\sqrt[3]{8}} \\[1mm] 6&=\dfrac{4k}{2} \\[1mm] 3&=k \end{align} $x=\dfrac{3{y}^{2}}{\sqrt[3]{z}}$ \begin{align}x&=\dfrac{3{\left(1\right)}^{2}}{\sqrt[3]{27}} \\[1mm] &=1 \end{align} Answer: [latex-display]x=20[/latex-display] ## Licenses & Attributions Cc licensed content, original. • Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution . ## CC licensed content, Shared previously • Question ID 91393,91394. Authored by: Jenck,Michael (for Lumen Learning). License: CC BY: Attribution . License terms: IMathAS Community License CC-BY + GPL. • College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution . License terms: Download for free at http://cnx.org/contents/ [email protected] . • Inverse Variation. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution . • Joint Variation: Determine the Variation Constant (Volume of a Cone). Provided by: Joint Variation: Determine the Variation Constant (Volume of a Cone) Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution . Please add a message. Message received. Thanks for the feedback. ## ORIGINAL RESEARCH article Joint problem-solving orientation, mutual value recognition, and performance in fluid teamwork environments. • 1 Harvard TH Chan School of Public Health, Harvard University, Cambridge, MA, United States • 2 School of Public Health, University of Texas Health Science Center at Houston, Houston, TX, United States Introduction: Joint problem-solving orientation (JPS) has been identified as a factor that promotes performance in fluid teamwork, but research on this factor remains nascent. This study pushes the frontier of understanding about JPS in fluid teamwork environments by applying the concept to within-organization work and exploring its relationships with performance, mutual value recognition (MVR), and expertise variety (EV). Methods: This is a longitudinal, survey-based field study within a large United States healthcare organization n = 26,319 (2019 response rate = 87%, 2021 response rate = 80%). The analytic sample represents 1,608 departmental units in both years (e.g., intensive care units and emergency departments). We focus on departmental units in distinct locations as the units within which fluid teamwork occurs in the hospital system setting. Within these units, we measure JPS in 2019 and MVR in 2021, and we capture EV by unit using a count of the number of disciplines present. For a performance measure, we draw on the industry-used measurement of perceived care quality and safety. We conduct moderated mediation analysis testing (1) the main effect of JPS on performance, (2) mediation through MVR, and (3) EV as a moderator. Results: Our results affirm a moderated mediation model wherein JPS enhances performance, both directly and through MVR; EV serves as a moderator in the JPS-MVR relationship. JPS positively influences MVR, irrespective of whether EV is high or low. When JPS is lower, greater EV is associated with lower MVR, whereas amid high JPS, greater EV is associated with higher MVR, as compared to lower EV. Discussion: Our findings lend further evidence to the value of JPS in fluid teamwork environments for enabling performance, and we document for the first time its relevance for within-organization work. Our results suggest that one vital pathway for JPS to improve performance is through enhancing recognition of the value that others offer, especially in environments where expertise variety is high. ## Introduction In today’s specialized and fast-paced world, organizations increasingly rely on fluid teamwork. Individuals often come together quickly and change frequently based on the needs of the organization or the nature of the task at hand ( Bushe and Chu, 2011 ; Li and van Knippenberg, 2021 ). This is common in industries from engineering to healthcare, where networks of diverse experts must be drawn upon to accomplish complex work in the moment ( Burke et al., 2004 ; Edmondson and Nembhard, 2009 ; Retelny et al., 2014 ). Teamwork in these settings can offer advantages of expertise pooling, knowledge integration, and shared accountability ( Cummings, 2004 ; Dahlin et al., 2005 ). Doing so fluidly may enable more efficient and adaptive use of expertise than stable team membership because individuals with distinct expertise can rapidly come and go as the need for their input arises or dissipates. This helps to address issues like urgency ( Klein et al., 2006 ), complexity ( Huckman et al., 2009 ), schedule shifts ( Valentine and Edmondson, 2015 ), and surprises ( Bechky and Okhuysen, 2011 ). However, fluid teamwork environments present challenges for organizational leaders who must establish conditions that enable effective teamwork, calling for new research to identify and understand the factors that are supportive ( Salas et al., 2018 ; Kerrissey et al., 2020 ). Teams are groups of individuals who interact to pursue a common goal ( Salas et al., 2008a ). Richard Hackman described “real” teams as teams that have stable and bounded membership, such that it is clear who is on the team and membership does not shift dramatically over time ( Hackman, 2002 ). Research has since identified stable teams as being advantageous for performance, suggesting that stable teams’ members gain a familiarity over time that confers a better understanding of one another’s strengths, weaknesses, backgrounds, and habits, which can stimulate both cognitive and social benefits ( Muskat et al., 2022 ). For this reason, it is often exhorted that teams be designed to remain relatively stable in order to derive the benefits of familiarity for performance. However, scholars over the past decade have noted that many dynamic work settings make stable and fixed team membership hard to achieve ( Edmondson, 2012 ; Tannenbaum et al., 2012 ; Mortensen and Haas, 2018 ; Li and van Knippenberg, 2021 ). Fluidity has been described as the presence of shifting team members, i.e., individuals moving on or off the team, though the term at times is also used to refer to ad hoc or short-duration teams ( Huckman and Staats, 2011 ; Dibble and Gibson, 2018 ). For clarity, we use fluidity to refer to membership change and “short duration” to refer to brief team lifespans (though note that, in real world settings, fluidity and short duration often overlap considerably). At the extreme of fluidity in teamwork, individuals may team up together in pursuit of shared goals on the fly or with such a short duration or high degree of individual turnover that ongoing familiarity as a coherent unit becomes elusive—what has been called teaming ( Edmondson, 2012 ) or dynamic participation ( Mortensen and Haas, 2018 ). Fluid teamwork can offer advantages in adaptiveness and efficiency because individuals are able to come and go as their contributions to a task or goal are needed, but it can also undermine familiarity and its potential benefits to teamwork. For example, it can limit the development of shared mental models and cohesion, which ordinarily help individuals to see joint work similarly and help them to depend upon and reciprocate with one another ( Bushe and Chu, 2011 ). The challenges of fluid teamwork are especially notable in the presence of varied expertise, as familiarity is important for bridging the knowledge differences that separate experts ( Kerrissey et al., 2021 ). In such circumstances, behaviors and orientations to collaboration may be especially valuable because they set expectations about whether and how to team up with others to share information, coordinate, and pursue overlapping goals, even when the structural conditions for ongoing, stable teamwork are not present ( Edmondson and Harvey, 2017 ). Amid calls for research on fluid teamwork for years ( Cronin et al., 2011 ; Wageman et al., 2012 ; Dibble and Gibson, 2018 ), there is a particular need for research that explores the contexts in which highly fluid teamwork transpires and that identifies factors that aid performance in the face of considerable barriers. In this study, we explore the unit conditions that may enable fluid teamwork to thrive, focusing on units within a context known for highly fluid teamwork: health care delivery. Many have written about the challenges of fluid teamwork in health care ( Bushe and Chu, 2011 ), such as shifting task needs due to the emergent nature of many health conditions, the presence of multi-disciplinarity (and its increase with expansion of medical expertise and the addition of new allied health roles), patient-centered frameworks that build unique clinician and staff teams around each patient’s needs, and increasing policy emphasis on team-based care ( Andreatta, 2010 ; Bedwell et al., 2012 ; Kerrissey et al., 2023 ). We focus on hospital-based care across units, such as emergency departments, medical intensive care units, surgical intensive care units, and transplant units, because of the common occurrence of shifting sets of individuals teaming up in service of a specific patient during their stay at the hospital. For instance, one study found that the average patient sees 17.8 professionals during a hospitalization (with a range of 5–44), and these individuals come and go throughout the stay as needed and available ( Whitt et al., 2007 ). Past research in such settings has detailed how fluid teamwork manifests. For example, teamwork shifts around patients in the emergency department with rotating work schedules as nurses and attending physicians clock in and out or specific consulting expertise is brought in for a unique need ( Valentine and Edmondson, 2015 ). Other research has described how intensive care units rely on shifting teamwork across core and peripheral members who may experience brief synchronous periods of work ( Mayo, 2022 ). Aligning with the perspective that health care teamwork often entails fluidity, we sought to examine organizational units (departments in distinct locations) to explore factors that managers and leaders may find useful in promoting effective teamwork in fluid settings and that would be measurable across a large number of work units in future research. Specifically, we focus on joint problem-solving orientations (JPS)—defined as emphasizing problems as shared and viewing solutions as requiring co-production—as a factor that has been found to promote performance in fluid teamwork settings and for which empirical and theoretical development remains nascent ( Kerrissey et al., 2021 ). Connecting with traditional research that has illuminated the value of shared orientations in more stable teams ( Driskell and Salas, 1992 ; Eby and Dobbins, 1997 ), the concept of JPS is especially relevant for fluid teamwork because it captures both the perceived jointness of the problem faced and the willingness to resolve it together, even without the luxury of stable team membership or fully aligned goals. Initial research on JPS was conducted in unique fluid teamwork settings—cross-sector, cross-organizational teams, and, later, in a computer simulation about a shopping task ( Kerrissey et al., 2021 ). Other research on fluid teams has been conducted in unique contexts, such as crowdsourced software coding ( Retelny et al., 2014 ). However, much fluid teamwork is more mundane, occurring within the bounds of organizations day-to-day, from software development ( Huckman et al., 2009 ) to health care delivery ( Bedwell et al., 2012 ). In these settings, establishing mechanisms and conditions that enable fluid teamwork to yield performance is vital for overcoming challenges and improving performance. This is especially important for work that relies on experts, who are known to face challenges in moving beyond their individual expertise to fully collaborate ( Reyes and Salas, 2019 ). However, we know little about both the mechanisms through which JPS affects performance and the boundary conditions that shape its effectiveness within organizational units where fluid teamwork is common. Drawing on multi-year field data from over 1,600 organizational units within a large, geographically-distributed healthcare organization, this paper makes two primary extensions. First, we apply JPS within work units in which fluid teamwork is common, examining its shared presence within organizational units and exploring its relationship with performance in this context. This perspective aligns with the conceptual claim that organizational environments affect teamwork ( Salas et al., 2018 ), alongside the practical reality that departments in hospitals are cogent entities that are used internally to structure work. This makes measurement of JPS within and across departments plausibly informative. Second, we test hypotheses about how JPS affects performance, proposing mutual value recognition (MVR) as a mediator and expertise variety (EV) as a moderator. We focus on MVR as describing the extent to which people recognize (i.e., respect, trust, and listen to) the value that others bring to collaboration. This may be vital for producing performance in fluid teamwork environments where diverse experts draw on distinct languages and norms ( Hall, 2005 ) and may face differences in near-term goals and commitments ( Bushe and Chu, 2011 ). In facilitating a shared focus on solving joint problems, JPS may allow individual experts to better and more rapidly recognize the value that others offer. Our specific hypotheses are detailed in the sections that follow. We build out a set of hypotheses to propose a moderated mediation model ( Figure 1 ), beginning with a main effect of JPS on performance, followed by a set of hypotheses pertaining to MVR as a mediator of that relationship. We conclude with moderation by expertise variety, hypothesizing that more variety heightens the positive relationship between JPS and both MVR and performance. Figure 1 . Hypothesized research model. ## Joint problem-solving orientation Past research has found that joint problem-solving orientation (JPS) is associated with improved work quality in fluid cross-boundary teams ( Kerrissey et al., 2021 ). This has two interrelated aspects: problem-solving and jointness. For problem-solving, seeking help with problem-solving tasks is central to knowledge-intensive work because it enables employees to address and complete complex tasks, thereby directly enhancing performance ( Hargadon and Bechky, 2006 ). The focus not only on problems but also on solving them further emphasizes the value of capturing the willingness and tendency for collaborators to move beyond venting ( Rosen et al., 2021 ) and toward solutions. The aspect of jointness, though related, is distinct, as individuals may seek help and advice for problem-solving, but that does not guarantee that they do so in a way that implies a shared sense of problem ownership among the asker and receiver of problem-solving assistance. The jointness aspect of JPS refers to this shared emphasis and understanding that problems are mutually faced and require solving together. Jointness is important because of a tendency toward separation among loosely affiliated people; for example, social categorization theory suggests that individuals tend to view others with shared goals, motivations, and priorities as the ingroup and to categorize those who do not as the outgroup ( Harrison and Klein, 2007 ). In related literature, establishment of collective orientation among team members, even in stable teams, has been identified as an important factor in team effectiveness ( Driskell and Salas, 1992 ; Eby and Dobbins, 1997 ; Hagemann and Kluge, 2017 ). For instance, research on computer-based simulations of complex teamwork tasks (e.g., extinguishing forest fires and protecting houses) found that, among a set of variables including trust and cohesion, only joint orientation of team members positively affected team performance ( Hagemann and Kluge, 2017 ). Other research on creative teams has found that teams that overcome asymmetries in psychological ownership of their ideas to generate collective ownership have more early successes ( Gray et al., 2020 ). In line with Kerrissey et al. (2021) , we posit that a joint orientation toward problem-solving is particularly relevant for fluid, knowledge-intensive teamwork contexts when diverse experts come together rapidly to solve problems. Kerrissey and co-authors examined this phenomenon in the extreme context of cross-sector, cross-organization teams that form ad hoc to solve pressing societal problems. We adapt their logic here to hypothesize that the relationship holds even in more ordinary work contexts (i.e., organizational units in health care). Our first hypothesis thus seeks to replicate the finding that JPS is positively associated with performance, but in the context of organizational work units where fluid teamwork occurs. H1: JPS is positively associated with performance. ## Mutual value recognition as a mediator Amid fluid teamwork, the need to swiftly establish a common understanding of what others offer becomes paramount ( Bushe and Chu, 2011 ), especially in the presence of expertise differences ( Reyes and Salas, 2019 ). Beyond a direct effect of JPS on performance through the concrete solving of organizational problems that would otherwise directly hinder performance, we hypothesize that the relationship of JPS to performance is also mediated through a greater recognition of the value that others in their environment offer. Extensive research shows that different expertise areas bring different values, perspectives and technical languages ( Carlile, 2004 ), including in health care ( Hall, 2005 ). Gaining familiarity with one another by working together over time can improve performance ( Huckman and Staats, 2011 ). However, in fluid expertise-driven work contexts where individuals fill roles in shifting sets based on their training (e.g., a nurse acting as a nurse across several teams, and being replaced by other nurses as needed), we posit that JPS enables people to better recognize the value in what other roles and expertise areas have to offer. In spurring problem and solution-focused collaborative work through shared recognition of problems as joint, JPS may help highly trained experts gain real-world experience with and respect for others’ work contributions. H2: JPS is positively associated with MVR. H3: MVR relates positively to performance. H4: MVR mediates the relationship between JPS and performance. ## EV as a moderator Expertise variety (EV) refers to heterogeneity among members of an interdependent work group who have each accumulated domain specific-knowledge, encompassing variations in functional role or educational background and skill ( Ericsson and Smith, 1991 ). On the one hand, the presence of varied expertise offers the advantage of a more heterogeneous pool of task-relevant perspectives and informational resources to draw from, which serves to enhance team performance ( Van Knippenberg et al., 2004 ). On the other hand, the presence of differing training or functional backgrounds can create communication and cooperation barriers and heighten relational conflicts, damaging interpersonal relationships and negatively affecting performance ( Cronin and Weingart, 2007 ). We hypothesize that EV in departments moderates the relationship of JPS with both MVR and performance. When there is high expertise variety within a department, we posit that the effect of JPS on performance and MVR is strengthened, as JPS can enable diverse experts to come together and mutually solve problems despite their differing backgrounds. When there is lower EV, we expect that JPS is still positively related to performance but less essentially so, as individuals with similar backgrounds may not need to rely on and value others to address problems collaboratively. Similarly, the benefit of JPS for performance that flows through MVR is likely especially important amid EV because the more experts present the more important it is likely to be that individuals value what others offer. H5: JPS in the presence of greater EV is related to greater MVR (H5a) and greater performance (H5b). H6: There is a moderated mediation that explains the relationship between JPS and performance, with MVR mediating the JPS-performance relationship and EV moderating the JPS-MVR and JPS-performance relationships. We collected data from a large, United States-based organization with over 20 hospitals, over 200 outpatient locations, and over 13 million patient encounters in 2022. It is commonly accepted that teamwork is central to most care delivery environments ( Rosenbaum, 2019 ) and that it is typically fluid ( Bedwell et al., 2012 ), in part because healthcare teams often engage varied expertise in response to patient needs ( Rosen et al., 2018 ). This makes a hospital-based healthcare organization an ideal setting for this study. ## Sample and administration The organizational survey was sent to 45,471 staff. We excluded individuals from our study who were in purely administrative departments to retain a focus on teamwork in patient-serving care, resulting in n = 26,319. The sample was composed of an array of expertise areas including patient-facing caregivers and their managers within the organization, which includes senior management, middle management, physicians, nurse practitioners, registered nurses, licensed practice nurses, nursing assistants, and other clinical professionals such as speech, physical and occupational therapists, alongside some security, service and clerical personnel supporting patient-facing departments. The survey was administered to staff electronically in English at two time points (May 2019 response rate = 87%; May 2021 response rate = 80%). The staff respondents were attributed to 1,608 departmental units, which were defined as being within the same department and physical location (in this organization, a single department can cut across several locations). These departmental units were obtained from human resources files. Table 1 presents the sample characteristics (presented for 2019). The study sample had a predominantly female composition (77.25%) and an age distribution with a large proportion in the 30–49 years age group (48.62%). A slight majority had 1–10 years of tenure (52.38%). There was a range of expertise areas present, with Registered Nurse being the most frequent (23.85%). Table 1 . Demographic statistics of the sample ( n = 26,319). All measures were assessed using five-point Likert scales and converted to domain means using the mean of the composite items. ## Joint problem solving Through iterative input sessions with organizational staff, we modified the joint problem-solving orientation measure developed by Kerrissey et al. (2021) for relevance within a single organization (the original measure was framed to ask about teamwork across two organizations). The adapted measure retained the theoretical emphasis of problems being seen as shared and solutions being seen as requiring co-production, but the language was modified to reflect departments as the referent unit. It included three items: (1) we view addressing problems as a team effort in this department, (2) when a problem arises, we routinely involve whomever is needed to address it, regardless of their unit or role, and (3) we can rely on people in other departments to address problems with us when needed, ( α = 0.85). JPS was measured in 2019. ## Mutual value recognition We measured MVR using relevant items from a validated survey developed for use in care delivery environments to capture affective teamwork across roles, the Primary Care Team Dynamics instrument, which includes 29 items all measured on Likert agreement scales and that are allocated across seven conceptual domains, including conditions for team effectiveness, shared understanding, accountability processes, communication processes, acting and feeling like a team, and perceived team effectiveness ( Song et al., 2015 , 2017 ). For the purpose of our hypothesizing in this study, we focused on the teamwork items used to capture valuing, trusting, and respecting others in expertise-diverse healthcare environments, which in the instrument’s measurement scheme fell under the broader theme of “acting and feeling like a team” (this theme also included two other aspects, one pertaining to using team skills and another on communicating information, which were not related to our hypothesizing and thus not measured in this study). In line with our interest in this study on mutually recognizing the value that others can offer, we focused on the three items describing aspects of valuing others, namely, respecting other roles and expertise, trusting each other’s work contributions, and listening to each other. MVR is thus distinct from the adjacent concept of transactive memory systems (TMS), which describes the shared division of cognitive labor in encoding, storage, retrieval, and communication of information ( Hollingshead, 2001 ). MVR focuses not on the cognitive representation and assignment of information from different domains but rather on the recognition that the information from other domains is valuable (respectable, trustable, and worth listening to). To measure this concept, we used the following items from Song et al. (2015) : (1) “People in this department show respect for each other’s roles and expertise,” (2) “People in this department trust each other’s work and contributions,” and (3) “Most of the time people in this department listen to the information that I communicate to them,” ( α = 0.86). We made slight updates to the original items to reflect the department as the referent entity rather than “team.” We used these items as measured in 2021 to mitigate common method bias concerns with their measurement alongside JPS. ## Outcome measure We captured performance in the context of healthcare delivery by measuring staff-perceived care quality and safety, leveraging practical measures used widely within the industry to inform operations and managerial decision-making. Specifically, we used measures from a survey conducted by the health system we studied through a national vendor (Press Ganey), which implements validated employee experience surveys in healthcare. Press Ganey’s industry-oriented research has found that employee perceptions derived from these surveys are related to patient ratings of care as well as hospital financial performance ( Buhlman and Lee, 2019 ). We draw on three measures from their survey, as captured in 2021: (1) “[This organization] provides high-quality care and service,” (2) “[This organization] makes every effort to deliver safe, error-free care to patients,” and (3) “I would recommend [this organization] to family and friends who need care.” These items are conceptually related as markers of performance in healthcare (i.e., that care is both high quality and safe, alongside the general measure of perceived performance based on likelihood of recommending their services to others); they were also empirically related with a high Cronbach’s alpha ( α = 0.92). For parsimony in presenting our results, we thus operationalize performance as a mean across the three interrelated items; sensitivity analyses examining each item separately yielded similar results. ## Aggregation of constructs Because we are interested in the organizational conditions in which fluid teamwork transpires, our measurement is within the department as the local environment that exists within physically located departments, with common management and workers who team up in shifting but overlapping configurations day after day. This approach has been used in prior research on psychological constructs, such as in the study of team climates using psychological safety, which has often been conducted at the departmental level in healthcare e.g., (see Nembhard and Edmondson, 2006 ). To justify this aggregation, we calculated within-team agreement parameters and intraclass correlations, and performed a one-way ANOVA for JPS, MVR, and team performance. All scales exhibited significant between-group variance ( F = 2.36, p < 0.01, F = 2.61; p < 0.01; and F = 3.28, p < 0.01, respectively). Intraclass correlations were: ICC 1 = 0.10 and ICC 2 = 0.57 for JPS; ICC 1 = 0.11 and ICC 2 = 0.62 for MVR; and ICC 1 = 0.11 and ICC 2 = 0.62 for performance. All scales showed moderate levels of agreement ( rwg = 0.70 for JPS; rwg = 0.71 for MVR; and rwg = 0.82 for performance). The ICC and rwg values were consistent with those in team research and considered acceptable for justifying aggregation ( Chen and Bliese, 2002 ; LeBreton and Senter, 2008 ). The typical values for ICC (1) are 0.01–0.45 and for ICC (2) are 0.45–0.90; values of rwg of 0.51–0.70 show moderate agreement, values of 0.71–0.90 show strong agreement, and 0.91–1.00 show very strong agreement ( LeBreton and Senter, 2008 ). ## Expertise variety Expertise variety was assessed as a sum of all professional/disciplinary title types in the department [job titles were provided through human resources records and were presented in a consistent fashion such that similar expertise and functional roles were labeled in the same way (e.g., Licensed Practice Nurse, Physician, etc.)]. This variable captures the variety of expertise and reflects the range of specialized knowledge and skills represented among individuals in each department. ## Control variable As a control measure, we included department size in our analysis, recognizing its established association with performance ( Salas et al., 2008b ). This was calculated as a sum of all individual people attached to a department-location in the human resources record. ## Analytic procedure We conducted CFA using structural equation modeling in Stata 15.1 for the individuals answering each item for JPS and MVR ( N = 24,563), examining root mean squared error of approximation (RMSEA), the chi-squared for the model vs. saturated, the Akaike’s information criterion (AIC), the comparative fit index (CFI), the Tucker Lewis Index (TLI), and the standardized root mean squared residuals (SRMR). We reviewed the descriptive means, standard deviations and correlations of the measures (as depicted in Table 2 ). Table 2 . Means, standard deviations, and bivariate correlations for the research variables. To test our hypotheses, we conducted a set of regression analyses using SPSS version 27, including a baseline regression, a mediation model, and moderated mediation using two models; we present the underlying regressions for these models in a stepwise fashion for clarity, in a series of Estimated Models (EM), which are each labeled within Table 3 . EM1 is a baseline model that includes the control variable of department size only. We then used the PROCESS macro for SPSS developed by Hayes (2018) to estimate the remaining models. To test hypotheses 1 through 4 pertaining to direct effects and mediation, we used a mediation model based on Hayes (2018) mediation “Model 4,” drawing on 5,000 random bootstrap samples. EM2 through EM4 in the results ( Table 3 ) build up the mediation model stepwise. Table 3 . Analytic results for baseline, mediation and moderated mediation models. For the moderated mediation analysis and testing of Hypotheses 5 and 6, we began with the Hayes mediation “Model 8,” which includes two moderating relationships for the moderator term (one with the mediator and one with the outcome). We used performance as the dependent variable, JPS as the independent variable, MVR as the mediator, and EV as the moderator (presented across EM5 and EM6 in Table 3 ). After finding that only one of the hypothesized moderating relationships was statistically significant (between EV and JPS with MVR), we then tested a moderated mediation model that used only that one moderating relationship (excluding the non-significant moderation between EV and JPS with performance) in order to check that the statistically significant moderated mediation holds with one moderating relationship between EV and JPS on MVR ( Hayes, 2018 ; “Model 7”). We present the findings from the moderated mediation with this single moderating relationship across EM5 and EM7. To interpret the form of the interactions in the moderated mediation analysis, we plotted the relationships between JPS, EV and performance/MVR using high and low levels of JPS and expertise at one standard deviation above and below their means ( Aiken et al., 1991 ). ## Confirmatory factor analysis Confirmatory factor analysis yielded a significant model with satisfactory goodness of fit ( Hu and Bentler, 1999 ; x 2 N = 24,563, p < 0.01, CFI = 0.997, TLI = 0.995, RMSEA = 0.038, SRMR = 0.016, AIC = 302377.974) suggesting that JPS and MVR loaded onto two factors as expected. The two-factor structure yielded a substantially better fit than when JPS and MVR were collapsed into one factor ( N = 24,563, p < 0.01, CFI = 0.701, TLI = 0.501, RMSEA = 0.361, SRMR = 0.182, AIC = 3309). These findings suggest that JPS and MVR are two distinct constructs ( Cangur and Ercan, 2015 ). ## Descriptive statistics Table 2 summarizes the descriptive statistics and the correlations among the control, independent, and dependent variables at the department level. ## Hypothesis testing Table 3 presents the results of the baseline, mediation, and moderated mediation models, all of which include department size as a control variable. Consistent with hypothesis 1, we found a significant relationship between JPS and performance ( b = 0.33, SE = 0.02, p < 0.01; see EM 2). We also found evidence consistent with hypothesis 2 relating JPS to MVR ( b = 0.62, SE = 0.02, p < 0.01; see EM 3). When regressing, JPS and MVR on performance (see EM 4), we found a significant relationship between JPS and performance ( b = 0.10, SE = 0.02, p < 0.01), and MVR and performance ( b = 0.37, SE = 0.02, p < 0.01). The effects of the mediation pathways are significant as follows: the total effect of JPS on performance = 0.33 (Bootstrapped SE = 0.02, with 95% CI [0.29, 0.36]), the direct effect of JPS on performance = 0.10 (Bootstrapped SE = 0.02, with 95% CI [0.06, 0.14]), and the indirect effect of JPS to performance through MVR is = 0.23 (Bootstrapped SE = 0.02, with 95% CI [0.20, 0.26]). Taken together, these results support Hypothesis 4 pertaining to the presence of mediation. For the first part of moderated mediation analysis, we regressed JPS, EV, and the interaction between JPS and EV on MVR (see EM 5). We found a significant interaction ( b = 0.05, SE = 0.02, p < 0.05), which supports hypothesis 5a. Figure 2A visually presents the form of the interaction, plotting the relationship between JPS, EV, and MVR using high and low levels of JPS and expertise at one standard deviation above and below their means ( Aiken et al., 1991 ). Figure 2 . (A) Two-way interaction between JPS and EV on MVR as the dependent variable. (B) Two-way interaction between JPS and EV on performance as the dependent variable. For the second part of moderated mediation model, we regressed JPS, MVR, and EV and the interaction between JPS and EV on performance (see EM 6). The interaction between JPS and EV was not statistically significant ( b = 0.02, SE = 0.01, p = 0.16). Hypotheses 5b and 6 were thus not supported. Figure 2B visually presents the form of the interaction, plotting the relationship between JPS and EV and performance using high and low levels of JPS and EV at one standard deviation above and below their means ( Aiken et al., 1991 ). We then re-tested the moderated mediation model while excluding the non-significant moderation between JPS and EV on performance from the model (Model 7, Hayes, 2018 ). We found a significant relationship between JPS and performance ( b = 0.10, SE = 0.02, p < 0.01; Table 2 , EM 7), and MVR and performance ( b = 0.37, SE = 0.02, p < 0.01). The index of moderated mediation (PROCESS, Model 7; Hayes, 2018 ) support a moderated mediation model (indirect effect = 0.02, Boot SE = 0.01, with 95% CI [0.01, 0.03]). The results support a moderated-mediation model with EV moderating the relationship between JPS and MVR, and MVR mediating the relationship between JPS and performance. This study sought to extend understanding of JPS within an organizational context characterized by fluid teamwork. We found evidence in support of a moderated mediation model, in which JPS was associated with performance directly and through MVR as a mediator, and in which JPS was most strongly related to MVR when expertise variety was high. These findings advance the nascent theory and research on JPS in fluid teamwork environments. They highlight JPS as valuable for organizations seeking to improve performance. Building upon the recently identified concept of JPS in research on fluid cross-sector teams ( Kerrissey et al., 2021 ), we found that JPS was also associated with performance within organizational work units that rely upon highly fluid teams of experts to conduct complex work. Our results show that this relationship held when controlling for departmental size, and the results indicate that a substantial proportion of the variance was explained even in the parsimonious models that we used (i.e., observing the r-squared terms ranging from 0.34 to 0.35). As an orientation, JPS is focused on the presence of a shared emphasis, focusing on the interpersonal rather than informational aspects of fluid teamwork—particularly, how people approach one another in reference to the work they are doing together and the problems they face. Though connected conceptually and likely empirically, it is thus distinct from other measures that focus on factors like transactive memory systems and information sharing (e.g., Hollingshead, 2001 ; Mesmer-Magnus and DeChurch, 2009 ). Our results lend evidence to JPS as a factor worth examining. Through mediation analysis, we found that part of the relationship between JPS and performance occurred through an enhanced recognition of the value that others can offer in collaborative work. This aligns with the perspective that experts must be able to swiftly establish a common understanding ( Reyes and Salas, 2019 )—and our findings lend evidence to the idea that JPS may help experts to do this more readily as they team up day-to-day with others. Put practically in an example, this represents the notion that a physician may not necessarily only need to learn afresh what a nurse “knows” but also to recognize that what a nurse knows about a patient from serving at their bedside is an important and valid input to the care process that is worth deliberately incorporating. This type of recognition may often come through familiarity in stable teams; it appears that JPS may also enable it, even without the luxury of stable teamwork over time. Our findings lend support to EV as a moderator. However, it was only statistically significant for the relationship between JPS and MVR and not for the direct relationship between JPS and performance. The pattern for the moderating relationship between JPS and MVR was notable. As Figure 2A depicts, we found that when JPS was low, high EV resulted in less MVR. As JPS increased, MVR increased for all levels of EV. When JPS was high, organizational units with more EV showed higher MVR than units with lower EV. This suggests that in units with less EV, a little JPS may go a long way to foster MVR, but when there is substantial EV, a relatively high amount of JPS may be needed to expand MVR. This underscores the importance of JPS in highly expertise-varied environments for rapidly establishing awareness of what other expertise domains can contribute. This is especially notable in contrast to the moderation of the direct relationship between JPS and performance, which though not statistically significant implied the potential of a notably different pattern, in which greater EV always strengthened the relationship between JPS and performance, regardless of the level of JPS ( Figure 2B ). This contrast seems plausible, as having more expertise to draw from extends the pool of task-relevant perspectives and informational resources to draw from, which serves to enhance team performance directly ( Van Knippenberg et al., 2004 ). Our findings suggest that for MVR this advantage is likely to differ, requiring substantial JPS to engender MVR when EV is high. ## Implications for theory and future research Our findings contribute to the emerging literature on fluid teamwork, for which there have been calls for more research ( Tannenbaum et al., 2012 ; Wageman et al., 2012 ; Mortensen and Haas, 2018 ). In exploring JPS within organizational units where highly fluid teamwork is dominant, we found that a factor that was initially studied in the unique context of cross-sector teams remained relevant, and our analyses enhanced our understanding of how it yields performance through the moderated mediation model we test. We nonetheless view our study with an exploratory lens, given the nascency of research on fluid teamwork and the empirical difficulty of studying fluid teamwork at scale within organizations, which often leads to “glimpses” rather than comprehensive pictures of this dynamic phenomenon ( Kerrissey et al., 2020 ). There is a great deal more to explore and learn. A main contribution of this research is to extend JPS to the organizational work unit context and to better understand how JPS operates and the boundary conditions that might shape its effectiveness in organizational contexts. While we find support for our conceptual model of moderated mediation, there are likely other important boundary conditions and mechanisms that can be proposed and explored in future research. For example, future research might examine how hierarchy and team climate measures such as psychological safety might relate to JPS and performance ( Nembhard and Edmondson, 2006 ). Research is also needed to identify ways to prompt JPS and to do so in a way that further facilitates MVR. One promising avenue may be through interventions focused on reflection; research on interprofessional collaboration, for instance, has underscored the value of reflection in helping individuals loosen the dominance of their tacitly acquired professional identities that prevent them from collaborating more effectively ( Wackerhausen, 2009 ). While we examine JPS and MVR across a 2-year timeframe, there is great opportunity to examine these relationships with more longitudinal time points and a greater focus on temporal developments. Both theory on teams and theory on problem solving present development as a temporal process; for example, the model of problem solving of Mac Duffie (1997) articulates problem definition, analysis, generation and selection of solutions, testing and evaluation of solutions, and routinization. Future theoretical work might integrate the temporal models around team development and problem-solving fruitfully to conceptualize how JPS unfolds. Further, there is opportunity for in-depth ethnographic research to examine JPS in real-world settings to identify its antecedents. This kind of in-depth longitudinal work may be particularly valuable given the likelihood of mutually reinforcing relationships; although our measurement timeframe and theory suggest that JPS generates MVR, it is also plausible that MVR further reinforces JPS. Indeed, the decision to ask for problem-solving assistance is enhanced by a cognitively-based appraisal of that person ( Nebus, 2006 ). Studies to investigate how team processes dynamically unfold are needed, for instance, event and time-based behavioral observation ( Kolbe and Boos, 2019 ). Because our purpose in this paper was to explore and extend the concept of JPS for fluid teamwork environments, we focused on JPS alongside a mediator and moderator rather than comparing JPS to alternative factors. We thus do not attempt to make a comprehensive model of team fluidity and performance—there are other factors that matter, and future research can explore how they compare to or interact with JPS. For example, our hypothesizing and findings in support of MVR as a key mediator differ from common explanations in more stable teamwork environments, where for instance collective psychological ownership is thought to be valuable for prompting effort, commitment and sacrifice among members, rather than elements of mutual value recognition ( Pierce and Jussila, 2011 ; Gray et al., 2020 ). It may be that in highly fluid teamwork among experts commitment mechanisms, though likely present to some degree, are less central because high fluidity may make commitment to any particular team entity less essential. This would be interesting to test in future research. A second, related area of extension could examine positive affective relationships, which are often cited as important factors in intact teams. However, research on problem-solving work has found a performance benefit to seeking out problem solving assistance from “dissonant ties,” i.e., difficult colleagues with whom a relationship may be fraught ( Brennecke, 2020 ). This points to potentially interesting and important differences in the role of positive affective bonds, relative to MVR, in highly fluid teamwork among experts; for instance, it is possible that MVR can develop effectively among dissonant ties, helping to value contributions even when other bonds remain suboptimal. Future research could compare and test these ideas. ## Implications for practice For practice, our findings suggest that organizational leaders and managers might look to joint problem-solving orientations as a key factor to promote performance within their organizational units where fluid teamwork occurs. This is important given that fluid teamwork is a reality in many highly dynamic, expertise-driven work settings ( Mortensen and Haas, 2018 ). Our research suggests that when fluid teamwork prevents people from gaining in-depth familiarity with other individuals—a key to performance in stable teams ( Hackman, 2002 )—they may nonetheless through joint problem-solving orientations come to better recognize the value of others’ contributions and thereby generate performance. For organizations, looking for ways to hire for, foster, measure and reward JPS may be highly valuable. Our measurement of JPS at the departmental level in this study suggests that organizations may use this level of measurement to inform and improve their fluid teamwork in practice, as they may find it onerous or infeasible to track such measures at the team level amid such high fluidity (e.g., in healthcare, it might otherwise require surveying staff for each of the many teams they interact with per day). Moreover, that a department-level measure of JPS has predictive power for performance in a fluid teamwork environment may also offer to practitioners a pragmatic entity for intervention. Consider the alternative for a highly dynamic organizational environment: even if an organization were able to collect data on each fluid team that formed, if those teams are so fluid, distinct and often short-lived as they are in healthcare, then it would nonetheless not be clear who would be responsible for intervening, when, or with whom. That organization might then have beautiful data on which teams perform best, and which need help, only to realize that none of the teams still exist. For this reason, a departmental or similar unit-based measurement approach may be advantageous to intervention within organizations. ## Limitations and future research This study has limitations. First, while the analysis was conducted across a large number of work units (i.e., departments), it was conducted within one overarching healthcare delivery organization. Future efforts to further test these measures and relationships in other organizations and industries where fluid teamwork is central are needed to inform the generalizability of our findings. There are aspects of healthcare delivery, such as the overarching shared mission of delivering high quality patient care across disciplines, that may make JPS more salient for performance and MVR more relevant as a mediator in this context than others. Second, there are limitations in measuring these concepts at a departmental level. We did not have access to measures of variation by degree of fluidity in teamwork, and presumably there is some variation in degree of fluidity across departments in our study that would be fruitful to explore. Moreover, we recognize that measuring constructs at the departmental unit is imperfect in healthcare, especially as additional teamwork occurs across departments. However, because the preponderance of work occurs within departments (i.e., within an intensive care unit) and because JPS influences the interactions among members teaming up within the department, we believe it is a useful and reasonable simplification. In addition, the consistency and agreement measures (e.g., ICCs and RWGs) we analyzed provided empirical evidence that the key constructs were similar within and different across the departments. Third, future research can further develop the concept and refine the measurement of MVR, given its significant relationship to JPS and performance in our exploratory analysis. Fourth, because we measured JPS and MVR within the department rather than measuring these factors in each fluid team that occurred, our results address the department environment rather than the fluid team as the unit of analysis. While this offers advantages for pragmatism and practice, it is also a limitation for understanding team-level orientations and processes. Future research could fruitfully extend theory in this area by observing JPS as it forms in the moment within fluid teams. ## Data availability statement The datasets presented in this article are not readily available because this is organizational data with sensitive employee information that authors have access to under condition of not sharing it. Requests to access the datasets should be directed to [email protected] . ## Ethics statement The study involving humans was approved by Harvard University Institutional Review Board. The studies were conducted in accordance with the local legislation and institutional requirements. Written informed consent for participation was not required from the participants or the participants’ legal guardians/next of kin in accordance with the national legislation and institutional requirements. ## Author contributions MK: Conceptualization, Data curation, Investigation, Methodology, Project administration, Writing – original draft, Writing – review & editing. ZN: Conceptualization, Data curation, Formal analysis, Methodology, Visualization, Writing – review & editing. 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The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. *Correspondence: Michaela Kerrissey, [email protected] Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher. #### COMMENTS 1. Joint Variation: Solving Joint Variation Problems in Algebra Example 1: Finding an Equation of Joint Variation Find an equation of variation where a varies jointly as b and c, and a = 30 when b = 2 and c =3. Solution Write the joint variation equation that resembles the general joint variation formula y = kxz. Let a = y, x = b, z = c. y = kxz a = kbc Recommended Caroline Astor: America's Society Queen 2. Solve problems involving joint variation For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable c, cost, varies jointly with the number of students, n, and the distance, d. A General Note: Joint Variation 3. Joint Variation - CCSS Math Answers Joint Variation - Formula, Examples | How to Solve Problems Involving Joint Variation? March 12, 2021 / By Prasanna Joint Variation definition, rules, methods and formulae are here. Check the joint variation problems and solutions to prepare for the exam. 4. Joint And Combined Variation Word Problems Solution: s ∝ d/t In other words, the longer the distance or the shorter the time, the faster is the speed. How To Solve Joint Variation Problems? Example: Suppose y varies jointly as x and z. What is y when x = 2 and z = 3, if y = 20 when x = 4 and z = 3? Show Video Lesson Joint Variation Problems Example: 5. Study Guide Solution Begin by writing an equation to show the relationship between the variables. x=\frac {k {y}^ {2}} {\sqrt [3] {z}} x = 3zky2 Substitute x = 6, y = 2, and z = 8 to find the value of the constant k. \begin {cases}6=\frac {k {2}^ {2}} {\sqrt [3] {8}}\hfill \\ 6=\frac {4k} {2}\hfill \\ 3=k\hfill \end {cases} ⎩⎨⎧6 = 38k22 6 = 24k 3 = k 6. Joint Variation Solution: The equation for the given problem of joint variation is x = Kyz where K is the constant. For the given data 16 = K × 4 × 6 or, K = 46 4 6. So substituting the value of K the equation becomes x = 4yz 6 4 y z 6 Now for the required condition x = 4×8×126 4 × 8 × 12 6 = 64 Hence the value of x will be 64. 2. 7. 2.7 Variation Word Problems Example 2.7.1 Find the variation equation described as follows: The surface area of a square surface (A) ( A) is directly proportional to the square of either side (x). ( x). Solution: 8. Joint Variation: Definition & Examples Example 1 - Abstract Let's look at some examples of joint variation problems. If y varies jointly with x, z, and w, and the value of y is 60 when x = 2, z = 3, and w = 5, what is the... 9. 1.8: Variation Solving Problems involving Direct, Inverse, and Joint variation ... Joint variation is a relationship in which one quantity is proportional to the product of two or more quantities. Combined variation exists when combinations of direct and/or inverse variation occurs . Example $$\PageIndex{3}$$: Joint Variation. The area of an ellipse varies ... 10. Joint and Combined Variation Combined variation is a mix of direct and indirect variation. The joint variation equation is z = k x m y n where k ≠ 0 and m > 0, n > 0. Review. For questions 1-5, write an equation that represents relationship between the variables. w varies inversely with respect to x and y. r varies inversely with the square of q. z varies jointly with x ... 11. Inverse and Joint Variation Example: Solving an Inverse Variation Problem A quantity [Math Processing Error] y varies inversely with the cube of [Math Processing Error] x. If [Math Processing Error] y = 25 when [Math Processing Error] x = 2, find [Math Processing Error] y when [Math Processing Error] x is 6. Show Solution Try It 12. Joint or Combined Variation (video lessons, examples and solutions) Joint Variation Examples Example: Suppose y varies jointly as x and z. What is y when x = 2 and z = 3, if y = 20 when x = 4 and z = 3? Show Video Lesson Example: z varies jointly with x and y. When x = 3, y = 8, z = 6. Find z, when x = 6 and y = 4. Show Video Lesson Joint Variation Application Example: 13. Direct, Inverse, Joint and Combined Variation Beginning Algebra Direct, Inverse, Joint and Combined Variation Direct, Inverse, Joint and Combined Variation When you start studying algebra, you will also study how two (or more) variables can relate to each other specifically. The cases you'll study are: Direct Variation, where one variable is a constant multiple of another. 14. CK12-Foundation This type of variation involves three variables, usually x, y and z. For example, in geometry, the volume of a cylinder varies jointly with the square of the radius and the height. In this equation the constant of variation is π, so we have V = π r 2 h. In general, the joint variation equation is z = k x y. Solving for k, we also have k = z x y. 15. 3.10: Modeling Using Variation Solving Problems Involving Joint Variation. Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation. For example, the cost of busing ... 16. Joint Variation Word Problems JOINT VARIATION WORD PROBLEMS Problem 1 : z varies directly with the sum of squares of x and y. z = 5 when x = 3 and y = 4. Find the value of z when x = 2 and y = 4. Solution : Since z varies directly with the sum of squares of x and y, z ∝ x2 + x2 z = k (x2 + y2) ---- (1) Substitute z = 5, x = 3 and y = 4 to find the value k. 5 = k (32 + 42) 17. Variation We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be inversely proportional and each term varies inversely with the other. Inversely proportional relationships are also called inverse variations.. For our example, the graph depicts the inverse variation.We say the water temperature varies inversely with ... 18. Joint Variation Examples and Word Problems In joint variation one variable is jointly proportional or jointly varies to two or more va... This is a video about Joint Variation Examples and Word Problems. 19. Applications and Variation Solving Problems involving Direct, Inverse, and Joint variation. Many real-world problems encountered in the sciences involve two types of functional relationships. The first type can be explored using the fact that the distance s in feet an object falls from rest, without regard to air resistance, can be approximated using the following ... 20. Joint Variation Definition Solving Joint Variation Problems. ... For example, if a problem states that z varies jointly with x and y, and z equals 12 when x is 2 and y is 3, we can find k by substituting these values into the formula: 12 = k(2)(3) Solving for k, we get: k = 2. Now, with the constant k known, we can predict z for any other values of x and y. Conclusion. 21. Exploring Combined Variation Examples and Joint Variation ... N=k b/p Here k is the constant of variation and cannot be 0. With an advertising budget of1200 and the t-shirt price of \$4.80, 6500 t-shirts are sold. Using this information, the value of k can be found. To do so, substitute N= 6500, b= 1200, and p= 4.80 in the above equation. N=kb/p. 6500=k ( 1200)/4.80.

22. 2.10: Modeling Using Variation

Identify the input, x ,and the output, y. Determine the constant of variation. You may need to divide y by the specified power of x to determine the constant of variation. Use the constant of variation to write an equation for the relationship. Substitute known values into the equation to find the unknown.

23. Study Guide

Example: Solving an Inverse Variation Problem. A quantity y y varies inversely with the cube of x x. If y=25 y = 25 when x=2 x = 2, find y y when x x is 6. Answer: The general formula for inverse variation with a cube is y=\dfrac {k} { {x}^ {3}} y = x3k. The constant can be found by multiplying y y by the cube of x x .

24. Joint problem-solving orientation, mutual value recognition, and

Joint problem solving. Through iterative input sessions with organizational staff, we modified the joint problem-solving orientation measure developed by Kerrissey et al. (2021) for relevance within a single organization (the original measure was framed to ask about teamwork across two organizations). The adapted measure retained the ...