Answer
a) $y= \frac{10935}{x^3}$
b) $y= 87.48$
Work Step by Step
Since $y$ varies inversely with $x^3$, we can write \begin{equation}
y= \frac{k}{x^3},
\end{equation} where $k$ is the constant of proportionality that must be determined.
a) Given that $y= 405$ when $x= 3$, we can use this information to find $k$ as follows: \begin{equation}
\begin{aligned}
405&= \frac{k}{3^3}\\
405\cdot 27&=k\\
\therefore k&=10935.
\end{aligned}
\end{equation} The required equation is
\begin{equation}
y= \frac{10935}{x^3}.
\end{equation} b) Determine $y$ when $x=5$.
\begin{equation}
y= \frac{10935}{5^3}= 87.48.
\end{equation} The answer is $y= 87.48$ when $x= 5$.
