Answer
a) $C(r)=6.28 r$
b) $75.36$ inches
c) $1.27$ inches
Work Step by Step
a) Let $C(r)$ be the circumference of the circle and $r$ be its radius. Since the circumference varies directly with the radius, we may write
\begin{equation}
\begin{aligned}
C(r)&= kr,
\end{aligned}
\end{equation} where $k$ is the constant of proportionality that must be determined. Given that $C(5)= 31.4$ inches when $r=5$ inches, we can use this information to find $k$ as follows:
\begin{equation}
\begin{aligned}
31.4&= 5k\\
\frac{31.4}{5}&= k\\
6.28&= k.
\end{aligned}
\end{equation} We can write
\begin{equation}
\begin{aligned}
C(r)&= 6.28r.
\end{aligned}
\end{equation} b) Find $C(12)$ when $r= 12$.
\begin{equation}
\begin{aligned}
C(r)&=6.28\cdot 12\\
&=75.36.
\end{aligned}
\end{equation} The circumference of a circle with a radius of $12$ inches is $75.36$ inches.
c) Set $C(r) = 8$ and solve for $r$.
\begin{equation}
\begin{aligned}
C(r)&=12\\
6.28r &=8\\
r&= \frac{8}{6.28}\\
& \approx 1.27.
\end{aligned}
\end{equation} The radius of a circle with a circumference of $8$ inches is about $1.27$ inches.
