Answer
a) $C(n)=150n$
b) $\$1050$
c) 5 nights
Work Step by Step
a) Let $C(n)$ be the cost for a hotel room per night and $n$ be the number of nights. Because the cost of hotel varies directly with the number of night, we may write
\begin{equation}
\begin{aligned}
C(n)&= kn,
\end{aligned}
\end{equation} vwhere $k$ is the constant of proportionality that must be determined. Given that $C(n)= 450$ when $n=3$, we can use this information to find $k$ as follows:
\begin{equation}
\begin{aligned}450&= 3k\\ \frac{450}{3}&= k\\
150&= k\\C(n)&=150 n.
\end{aligned}
\end{equation} b) Find $C(7)$ when $n= 7$
\begin{equation}
\begin{aligned}
C(7)&=150\cdot 7\\
&=1050.
\end{aligned}
\end{equation} The cost of renting hotel for $7$ nights is $\$1050$.
c) Set $C(n) = 800$ and solve for $n$. \begin{equation}
\begin{aligned}
C(n)&=150k\\
150k &=800\\
k&= \frac{800}{150}\\
& \approx5.33.
\end{aligned}
\end{equation} You can stay for $5$ nights at the hotel with a budget of $\$800$.
