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

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Identify the type of variation in the problem. Since x varies directly with the cube of z and inversely with y, you can express this relationship using the formula: x = k \frac{z^3}{y}, where k is the constant of proportionality.

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 eliminate the fraction: xy = k z^3.

Next, divide both sides of the equation by x to solve for y: y = \frac{k z^3}{x}.

This equation expresses y in terms of x, z, and the constant k. It shows that y is directly proportional to the cube of z and inversely proportional to x.

To find specific values of y, substitute known values of x, z, and k into the equation y = \frac{k z^3}{x}.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 of z' means that x = k * z^3 for some constant k. This concept is essential for establishing the initial equation based on the relationship between x and z.

Inverse Variation

Inverse variation occurs when one variable increases as another decreases, typically expressed as y = k/x. In the context of the question, 'x varies inversely as y' indicates that x = k/y, which is crucial for incorporating the relationship between x and y into the equation.

Combining Direct and Inverse Variation

When a variable varies directly and inversely with respect to others, the relationships can be combined into a single equation. For this problem, we can express x as a function of both z and y, leading to the equation x = k * z^3 / y. Solving this equation for y will provide the necessary relationship to understand how y depends on x and z.

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