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Chapter 7: Problem 49

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

Short Answer

Expert verified
The given equation \( z=\frac{k\sqrt{x}}{y^{2}} \) exhibits a combined direct and inverse variation. Specifically, z varies directly as the square root of x, and inversely as the square of y. As x increases, z increases, and as y increases, z decreases.

Step by step solution

01

Interpretation of Equation

The equation \( z=\frac{k\sqrt{x}}{y^{2}} \) can be broken down as follows: z varies directly as the square root of x (since z and \(\sqrt{x}\) are in the numerator) and inversely as the square of y (since y is in the denominator). The constant of variation, often denoted as k, is still present, which is a characteristic of variation problems.
02

Identifying the Types of Variation

From the equation, it can be observed that it represents both a direct and inverse variation. A direct variation exists when one variable increases or decreases directly with the other, which here is z and \(\sqrt{x}\). An inverse variation is when an increase in one variable corresponds to a decrease in the other, which here is z and \(y^{2}\). Hence, in simple terms, when x increases, z increases and when y increases, z decreases.
03

Conclusion

From the given equation \( z=\frac{k\sqrt{x}}{y^{2}} \), it can be concluded that z varies directly as the square root of x and inversely as the square of y. This means that z increases with increasing x (because the square root of x increases with x) and z decreases with increasing y (because 1 over the square of y decreases with increasing y).

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