Answer
a) \begin{equation}
\begin{aligned}
y &= \frac{200\sqrt{6}}{8\sqrt{x}}\\
\end{aligned}
\end{equation}
b) The answer is $y=30.62$ when $x= 4$.
Work Step by Step
Since $y$ varies inversely with $8\sqrt{x}$, we can write \begin{equation}
y= \frac{k}{8\sqrt{x}},
\end{equation} where $k$ is the constant of proportionality that must be determined.
a) Given that $y= 25$ when $x= 6$, we can use this information to find $k$ as follows: \begin{equation}
\begin{aligned}
25&= \frac{k}{8\sqrt{6}}\\
25\cdot 8\sqrt{6}&=k\\
\therefore k&=200\sqrt{6}.
\end{aligned}
\end{equation} The required equation is
\begin{equation}
\begin{aligned}
y &= \frac{200\sqrt{6}}{8\sqrt{x}}.
\end{aligned}
\end{equation} b) Determine $y$ when $x=4$:
\begin{equation}
\begin{aligned}
y &= \frac{200\sqrt{6}}{8\sqrt{4}}\\
& \approx 30.62.
\end{aligned}
\end{equation} The answer is $y=30.62$ when $x= 4$.
