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

Verified step by step guidance

1. Identify the type of variation in the equation: The equation z = k√x / y^2 suggests a combined variation where z varies directly with the square root of x and inversely with the square of y.
2. Express the direct variation: The term √x indicates that z increases as the square root of x increases, assuming all other variables are held constant.
3. Express the inverse variation: The term y^2 in the denominator indicates that z decreases as y increases, assuming all other variables are held constant.
4. Understand the constant k: The constant k in the equation is the proportionality constant that adjusts the rate of variation between z, x, and y.
5. Combine the variations: Overall, the equation shows that z is proportional to the square root of x and inversely proportional to the square of y, scaled by the constant k.

Key Concepts

Direct Variation

Direct variation occurs when one variable is a constant multiple of another. In the context of the equation z = k√x / y², z varies directly with the square root of x, meaning that as x increases, z increases proportionally, provided y remains constant.

Inverse Variation

Inverse variation describes a relationship where one variable increases as another decreases. In this equation, z varies inversely with y², indicating that as y increases, z decreases, assuming x remains constant. This relationship highlights how changes in one variable can affect another in opposite ways.

Constant of Variation

The constant of variation, represented by k in the equation, is a fixed value that determines the strength of the relationship between the variables. It scales the relationship between z, x, and y², indicating how much z changes in response to changes in x and y, thus providing a specific context for the variation described.