# Inverse and Joint Variation

### Learning Objectives

- Solve an Inverse variation problem
- Write a formula for an inversely proportional relationship

d, depth |
[latex]T=\frac{\text{14,000}}{d}[/latex] | Interpretation |
---|---|---|

500 ft | [latex]\frac{14,000}{500}=28[/latex] | At a depth of 500 ft, the water temperature is 28° F. |

350 ft | [latex]\frac{14,000}{350}=40[/latex] | At a depth of 350 ft, the water temperature is 40° F. |

250 ft | [latex]\frac{14,000}{250}=56[/latex] | At a depth of 250 ft, the water temperature is 56° F. |

**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 [latex]y=\frac{k}{x}[/latex] 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 [latex-display]y=\frac{k}{{x}^{n}}[/latex-display] 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 [latex]k={x}^{n}y[/latex].

### 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.Answer:
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.

[latex]\begin{array}{c}t\left(v\right)=\frac{100}{v}\hfill \\ \text{ }=100{v}^{-1}\hfill \end{array}[/latex]

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.**
**

- 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

A quantity*y*varies inversely with the cube of

*x*. If

*y*= 25 when

*x*= 2, find

*y*when

*x*is 6.

Answer:
The general formula for inverse variation with a cube is [latex]y=\frac{k}{{x}^{3}}[/latex]. The constant can be found by multiplying *y* by the cube of *x*.

[latex]\begin{array}{c}k={x}^{3}y\hfill \\ \text{ }={2}^{3}\cdot 25\hfill \\ \text{ }=200\hfill \end{array}[/latex]

Now we use the constant to write an equation that represents this relationship.[latex]\begin{array}{c}y=\frac{k}{{x}^{3}},k=200\hfill \\ y=\frac{200}{{x}^{3}}\hfill \end{array}[/latex]

Substitute*x*= 6 and solve for

*y*.

[latex]\begin{array}{c}y=\frac{200}{{6}^{3}}\hfill \\ \text{ }=\frac{25}{27}\hfill \end{array}[/latex]

#### Analysis of the Solution

The graph of this equation is a rational function.### Try It

A quantity*y*varies inversely with the square of

*x*. If

*y*= 8 when

*x*= 3, find

*y*when

*x*is 4.

Answer: [latex]\frac{9}{2}[/latex]

## 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 [latex]x=\frac{ky}{z}[/latex]. 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.

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

[latex]x=\frac{k{y}^{2}}{\sqrt[3]{z}}[/latex]

Substitute*x*= 6,

*y*= 2, and

*z*= 8 to find the value of the constant

*k*.

[latex]\begin{array}{c}6=\frac{k{2}^{2}}{\sqrt[3]{8}}\hfill \\ 6=\frac{4k}{2}\hfill \\ 3=k\hfill \end{array}[/latex]

Now we can substitute the value of the constant into the equation for the relationship.[latex]x=\frac{3{y}^{2}}{\sqrt[3]{z}}[/latex]

To find*x*when

*y*= 1 and

*z*= 27, we will substitute values for

*y*and

*z*into our equation.

[latex]\begin{array}{c}x=\frac{3{\left(1\right)}^{2}}{\sqrt[3]{27}}\hfill \\ \text{ }=1\hfill \end{array}[/latex]

### Try It

*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.

Answer: [latex]x=20[/latex]

## 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.