Explanation:
Here we need to write a function that models each variation, so we know that:
[tex]x = \frac{1}{3} \\ \\ y = \frac{1}{5} \\ \\ r = 3 \\ \\ z = \frac{1}{2}[/tex]
We also know that z varies directly with x and inversely with the product of r² and y. In other words, we can write z as follows:
[tex]z=k\frac{x}{r^2y} \\ \\ \\ \text{Where k is a real constant.} \\ \\ \\ \text{By substituting x,y,r and z into the equation we can get k:} \\ \\ \\ \frac{1}{2}=k\frac{\frac{1}{3}}{3^2(\frac{1}{5})^2} \\ \\ \frac{1}{2}=k\frac{\frac{1}{3}}{\frac{9}{25}} \\ \\ \frac{1}{2}=k(\frac{25}{27}) \\ \\ k=\frac{27}{50}[/tex]
Therefore, the model is:
[tex]\boxed{z=\frac{27}{50}\frac{x}{r^2y}}[/tex]
