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 of $z$ and inversely as the square root of $w$ and of $t .$ When $x=2, z=3,$ $w=16,$ and $t=3,$ then $y=1 .$ Find $y$ when $x=3, z=2, w=36,$ and $t=5 .$

Key Concepts

Inverse Variation

Inverse variation indicates that one variable increases as the other decreases in such a way that their product is constant. In problems involving joint variation, certain variables may affect the outcome inversely, and this is reflected in the equation by having the variable of interest inversely proportional to some function of the other variables.

Constant of Proportionality

The constant of proportionality is a fixed number that relates the variables in both direct and inverse variations. It allows one to scale the relationship accurately so that when the independent variables are known, the dependent variable can be calculated. Finding this constant is a crucial step in solving problems involving joint, direct, and inverse variations.

Joint Variation

Joint variation occurs when a variable depends on two or more other variables either directly or indirectly, often through a combination of these relationships. It involves setting up an equation where the variable is equal to a constant multiplied by the product or quotient of the given variables raised to specified powers.

Direct Variation

Direct variation means that one variable increases or decreases in a constant proportion to another. This relationship is typically expressed with an equation where the variable of interest is equal to a constant times the other variable. In the context of joint variation, some factors directly affect the outcome through this type of proportionality.

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