Water temperature in an ocean varies inversely to the water’s depth. Between the depths of 250 feet and 500 feet, the formula [latex]T=\frac{14,000}{d}[/latex] 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 [latex]y=\frac{k}{x}[/latex] for inverse variation in this case uses k = 14,000.

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.

Solution

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.

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.

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