Solve 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 [latex]x=\frac{ky}{z}[/latex]. Notice that we only use one constant in a joint variation equation.

Example 4: 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.

Solution

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{cases}6=\frac{k{2}^{2}}{\sqrt[3]{8}}\hfill \\ 6=\frac{4k}{2}\hfill \\ 3=k\hfill \end{cases}[/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{cases}x=\frac{3{\left(1\right)}^{2}}{\sqrt[3]{27}}\hfill \\ \text{ }=1\hfill \end{cases}[/latex]

Try It 3

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.

Solution