In Exercises 45–46, describe in words the variation shown by the given equation. z = kx^2 √y

Verified step by step guidance

1. Identify the variables and constants in the equation $z=k{x}^2\sqrt{y}$, where $z$ is the dependent variable, $x$ and $y$ are independent variables, and $k$ is a constant of proportionality.
2. Recognize that $z$ varies directly with the square of $x$, meaning if $x$ increases, $z$ increases proportionally to the square of $x$.
3. Understand that $z$ also varies directly with the square root of $y$, so if $y$ increases, $z$ increases proportionally to the square root of $y$.
4. Combine these observations to describe the overall variation: $z$ varies jointly as the square of $x$ and the square root of $y$.
5. Summarize the variation in words: $z$ increases as $x$ increases squared and as the square root of $y$ increases, scaled by the constant $k$.

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable increases or decreases proportionally with another. In the equation z = kx^2 √y, z varies directly with x squared and the square root of y, meaning as x or y increase, z changes accordingly based on the constant k.

Exponents and Powers

Exponents indicate repeated multiplication of a base number. Here, x is raised to the power of 2 (x^2), meaning x is multiplied by itself. Understanding how exponents affect variables helps interpret how changes in x influence z.

Square Root Function

The square root function, denoted √y, represents a value that when squared returns y. It grows slower than linear functions, so as y increases, √y increases but at a decreasing rate. Recognizing this helps explain how y affects z in the equation.