Answer
a) $r = \frac{C}{2\pi}$
b) $1.1$ ft, $1.3$ ft, $1.4$ ft
c) $\text{Area} = \frac{C^{2}}{4 \pi} $
Work Step by Step
a) The circumference $C$ of a circle with radius $r$ is $$C = 2\pi r.$$ Divide both sides by $2\pi$ $$\frac{C}{2\pi} = r.$$
Therefore the rewritten formula for $r$ is $$r = \frac{C}{2\pi}.\tag{1}$$
b) We substitute $C = 7$ in equation $(1)$ $$r = \frac{7}{2\pi} = 1.114... \approx 1.1\text{ ft}.$$
We substitute $C = 8$ in equation $(1)$ $$r = \frac{8}{2\pi} = 1.273... \approx 1.3\text{ ft}.$$
We substitute $C = 9$ in equation $(1)$ $$r = \frac{9}{2\pi} = 1.432... \approx 1.4\text{ ft}.$$
c) The area, $A$, of a cross section of a circle with radius $r$ is $$A = \pi r^{2}.\tag{2}$$ Since the circumference $C$ is known, we substitute $r$ from equation $(1)$ in equation $(2)$:
$$\begin{align}
A& = \pi \left(\frac{C}{2\pi}\right)^{2}\\
A& = \pi \left(\frac{C^{2}}{4 \pi^{2}}\right) \text{ (square the value in brackets)}\\
A& = \frac{C^{2}}{4 \pi} \text{ (simplify)}.
\end{align}$$ This equation can be used to derive the area of a cross section when the circumference is known.
