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.

Verified step by step guidance

1. Identify the type of variation in the problem. Since x varies directly with the cube root of z and inversely with y, we can express this relationship using the formula: x = k \cdot \frac{\sqrt[3]{z}}{y}, where k is the constant of proportionality.
2. Isolate y in the equation to solve for it in terms of x, z, and k. Start by multiplying both sides of the equation by y to get y \cdot x = k \cdot \sqrt[3]{z}.
3. Next, divide both sides of the equation by x to solve for y: y = \frac{k \cdot \sqrt[3]{z}}{x}.
4. This equation expresses y in terms of x, z, and the constant k. To find a specific value of y, values for x, z, and k must be known.
5. The final equation y = \frac{k \cdot \sqrt[3]{z}}{x} can be used to determine y given specific values for x, z, and k, or to analyze how changes in x and z affect 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.