Answer
a) $S(h)=12.5 h$
b) $\$510$
c) $31.37$ hours
Work Step by Step
a) Let $S(h)$ be Peter's weekly salary and $h$ be the total number of hours that he works per week. Because his salary varies directly with the number of hours, we may write
\begin{equation}
\begin{aligned}
S(h)&= kh,
\end{aligned}
\end{equation} where $k$ is the constant of proportionality that must be determined. Given that $S(h)= 382.5$ when $h=30$, we can use this information to find $k$ as follows: \begin{equation}
\begin{aligned}
382.5&= 30k\\
\frac{382.5}{30}&= k\\
12.75&= k\\
S(h)&= 12.5h
\end{aligned}
\end{equation} b) Find $S(40)$ when $h= 40$.
\begin{equation}
\begin{aligned}
S(40)&=12.5\cdot 40\\
&= 510.
\end{aligned}
\end{equation} Peter's salary for a $40$ hours work is $\$510$.
c) Set $S(h) = 400$ and solve for $h$.
\begin{equation}
\begin{aligned}
S(h)&=400\\
12.5h&=400\\
h&= \frac{400}{12.75}\\
& \approx 31.37
\end{aligned}
\end{equation} He would have to work about $31.37$ hours to earn $\$400$.
