Answer
Our function that models the relationship is:
$z = \frac{4}{xy}$
$z = \frac{1}{9}$ when $x = 4$ and $y = 9$.
Work Step by Step
The formula to describe a number $z$ that varies inversely with $x$ and $y$ is:
$z = \frac{k}{xy}$, where $k$ is the constant of variation.
Plug in the values given so we can find $k$:
$0.5 = \frac{k}{(2)(4)}$
Simplify:
$0.5 = \frac{k}{8}$
Multiply each side of the equation by $8$ to solve for $k$:
$k = 4$
Therefore, our function that models the relationship is:
$z = \frac{4}{xy}$
We are asked to find $z$ when $x$ is $4$ and $y$ is $9$:
$z = \frac{4}{(4)(9)}$
Cancel common factors in the numerator and denominator:
$z = \frac{1}{9}$
