Problem 46 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The variable z in the equation varies directly as the square of x and the square root of y.

Step 1: Explain the form of the equation

The given equation has the form of a direct variation equation which can be represented as \(z = k * f(x, y)\), where k is a constant of proportionality and f(x, y) is a function of x and y.

Step 2: Identify the variables' relationship with z

Here, it can be seen that z varies directly with \(x^{2}\), as x is squared in this function. This means that as x increases, z also increases by the square of the factor by which x increases. Similarly, z also varies directly with \(\sqrt{y}\), which means that as y increases, z increases by the square root of the factor by which y increases.

Step 3: Formulate the variation in words

The variation in the given equation can be described as follows: z varies directly with the square of x and the square root of y