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

Verified step by step guidance

1. Identify the variables and constants in the equation \(z = \frac{k \sqrt{x}}{y^2}\), where \(z\) is the dependent variable, \(x\) and \(y\) are independent variables, and \(k\) is a constant.
2. Recognize that \(z\) varies directly with the square root of \(x\), meaning as \(x\) increases, \(z\) increases proportionally to \(\sqrt{x}\).
3. Observe that \(z\) varies inversely with the square of \(y\), meaning as \(y\) increases, \(z\) decreases proportionally to \(\frac{1}{y^2}\).
4. Combine these observations to describe the overall variation: \(z\) increases with \(\sqrt{x}\) and decreases with \(y^2\).
5. Express the variation in words: \(z\) varies directly as the square root of \(x\) and inversely as the square of \(y\).

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable increases or decreases proportionally with another. In the equation z = k√x / y², z varies directly with the square root of x, meaning as x increases, z increases proportionally to √x, assuming other variables remain constant.

Inverse Variation

Inverse variation occurs when one variable increases as another decreases, typically expressed as a variable divided by another. Here, z varies inversely with y squared, indicating that as y increases, z decreases proportionally to 1/y², assuming other variables are constant.

Square Root and Exponent Rules

Understanding square roots and exponents is essential to interpret the equation. The square root of x (√x) is equivalent to x raised to the 1/2 power, and y squared (y²) means y multiplied by itself. These operations affect how changes in x and y influence z.