Problem 56 - College Algebra

Question

Describe in words the variation shown by the given equation. \(z=k x^{2} \sqrt{y}\)

Step-by-Step Solution

Answer

The equation \(z=k x^{2} \sqrt{y}\) shows a direct variation. Z varies directly as the square of x and the square root of y.

Step 1: Identify the type of variation

In this equation, \(z=k x^{2} \sqrt{y}\), if k is a constant, \(z \propto x^{2} \sqrt{y}\) holds. This implies that z varies directly as the square of x and the square root of y.

Step 2: Describe how z varies with respect to x

The equation \(z=k x^{2} \sqrt{y}\) shows z is directly proportional to \(x^{2}\). This implies that if x increases, z will grow in respect to the square of the increment of x, assuming y remains constant.

Step 3: Describe how z varies with respect to y

The equation \(z=k x^{2} \sqrt{y}\) shows that z is directly proportional to \(\sqrt{y}\). This implies that if y increases, z will grow in relation to the square root of the increment of y, assuming x remains constant. If y decreases under the same assumptions, z will shrink in relation to the square root of the decrement of y.