Answer
a) $y= \frac{1867776}{x^7}$
b) $y= 6.67$
Work Step by Step
Since $y$ varies inversely with $x^7$, we can write \begin{equation}
y= \frac{k}{x^7},
\end{equation} where $k$ is the constant of proportionality that must be determined.
a) Given that $y= 114$ when $x= 4$, we can use this information to find $k$ as follows: \begin{equation}
\begin{aligned}
117&= \frac{k}{4^7}\\
114\cdot 16384&=k\\
\therefore k&=1867776.
\end{aligned}
\end{equation} The required equation is \begin{equation}
y= \frac{1867776}{x^7}.
\end{equation} b) Determine $y$ when $x=6$.
\begin{equation}
y= \frac{1867776}{6^7}= 6.67.
\end{equation} The answer is $y= 6.67$ when $x= 6$.
