# Question 15

\begin{align} & \text{If }r\text{ varies inversely as the square root of }s\text{ and }t\text{ how does }s\text{ vary with }r\text{ and }t. \\ & (A)\text{ }s\text{ varies directly as }r\text{ and }t \\ & (B)\text{ }s\text{ varies inversely as }r\text{ and }{{t}^{2}} \\ & (C)\text{ }s\text{ varies inversely as }{{r}^{2}}\text{ and }t \\ & (D)\text{ }s\text{ varies directly as }{{r}^{2}}\text{ and }{{t}^{2}} \\\end{align}
\begin{align} & r\propto \frac{1}{\sqrt{st}} \\ & r=\frac{k}{\sqrt{st}}\text{ }(k=\text{ proportionality constant)} \\ & \text{Square both sides} \\ & {{r}^{2}}=\frac{{{k}^{2}}}{st} \\ & \text{multiply both sides by }s \\ & s{{r}^{2}}=\frac{{{k}^{2}}}{t} \\ & \text{Divide both sides by }{{r}^{2}} \\ & s=\frac{{{k}^{2}}}{{{r}^{2}}t}\text{ } \\ & \text{Let }{{k}^{2}}=A\text{ (where }A\text{ is a constant)} \\ & s=\frac{A}{{{r}^{2}}t} \\ & s\propto \frac{1}{{{r}^{2}}t} \\ & s\text{ varies inversely as the }{{r}^{2}}\text{ and }t \\\end{align}