In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies directly as the cube root of z and inversely as y.

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is a constant multiple of another. In this case, 'x varies directly as the cube root of z' means that if z increases, x increases proportionally, and can be expressed as x = k * (z^(1/3)), where k is a constant.

Inverse Variation

Inverse variation occurs when one variable increases as another decreases. The phrase 'inversely as y' indicates that as y increases, the value of x decreases. This relationship can be expressed as x = k' / y, where k' is another constant.

Combining Direct and Inverse Variation

When combining direct and inverse variations, we can express the relationship using both types of variation in a single equation. For the given problem, we can combine the two relationships to form an equation like x = k * (z^(1/3)) / y, which can then be solved for y to find its value in terms of x and z.