For the following exercises, use the given information to...

For the following exercises, use the given information to find the unknown value.
$y$ varies jointly as the square of $x$ and the cube of $z$ and inversely as the square root of $w .$ When $x=2, z=2,$ and $w=64,$ then $y=12 .$ Find $y$ when $x=1, z=3,$ and $w=4$ .

Key Concepts

Constant of Proportionality

This is a fixed number that links the variables in a variation relationship. When the relationship is expressed as a formula, the constant of proportionality is determined using known values. Once found, it can be applied to compute the dependent variable for different sets of independent variable values.

Joint Variation

This concept refers to a situation where a variable depends on two or more other variables, each raised to a specific power, and their product determines the value of the dependent variable. In a joint variation, the dependent variable changes with the combined effect of these independent variables.

Direct Variation

Direct variation describes a relationship where one variable increases or decreases in direct proportion to another variable. In the context of joint variation, parts of the product (such as the square of one variable or the cube of another) indicate that increases in those values will lead to a proportional increase in the solution of the dependent variable.

Inverse Variation

Inverse variation occurs when an increase in one variable causes a proportional decrease in another, typically represented by dividing by the variable or its function. In complex variation problems, combining direct and inverse variation means some factors increase the dependent variable while others decrease it.

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