Problem 55 - College Algebra

Question

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

Step-by-Step Solution

Answer

The function \(z=\frac{k \sqrt{x}}{y^{2}}\) varies inversely with the square of 'y' and varies directly with the square root of 'x'.

Step 1: Identifying the type of variation when ‘y’ changes

Firstly, consider y independent of x. Looking at the function \(z=\frac{k}{y^{2}}\) which is known as the inverse square variation. Here, as 'y' increases by a certain factor, 'z' decreases by the square of that factor, assuming all other variables are kept constant. Conversely, as 'y' decreases by a certain factor, 'z' increases by the square of that factor.

Step 2: Identifying the type of variation when ‘x’ changes

Secondly, consider x independent of y. Looking at the function \(z=k \sqrt{x}\), which is known as the direct square root variation. Here, when 'x' increases by a certain factor, 'z' increases by the square root of that factor, assuming all other variables are kept constant. Conversely, when 'x' decreases by a certain factor, 'z' decreases by the square root of that factor.

Step 3: Conclusively Describing the Variation

In conclusion, the function \(z=\frac{k \sqrt{x}}{y^{2}}\) is simultaneously inversely proportional to the square of 'y' and directly proportional to the square root of 'x'. Consequently, when 'x' increases, 'z' increases, and when 'y' increases, 'z' decreases. Additionally, the factor of increase or decrease for 'z' depends on the square root for 'x' and the square of 'y'.