Question 9
10 pts
One of the values $\mathrm{r}$ (radius), $\mathrm{d}$ (diameter), $\mathrm{V}$ (volume), or $\mathrm{S}$ (surface area) is given for a particular sphere. Find the indicated value. Leave $\pi$ in your answer.
\[
\mathrm{V}=\frac{1372}{3} \pi \mathrm{ft}^{3} ; \mathrm{S}=?
\]
$14 \pi \mathrm{ft}^{2}$
$196 \mathrm{ft}^{2}$
$196 \pi \mathrm{ft}^{2}$
$\frac{196}{3} \pi \mathrm{ft}^{2}$

Step 1 ：Given that the volume of the sphere is \(\frac{1372}{3} \pi \) cubic feet.

Step 2 ：The volume of a sphere is given by the formula \(V = \frac{4}{3} \pi r^3\).

Step 3 ：Setting the given volume equal to the formula, we get \(\frac{1372}{3} \pi = \frac{4}{3} \pi r^3\).

Step 4 ：Solving for \(r^3\), we divide both sides by \(\frac{4}{3} \pi\), yielding \(r^3 = \frac{1372}{4}\) cubic feet.

Step 5 ：Simplifying, we find that \(r^3 = 343\) cubic feet.

Step 6 ：Taking the cube root of both sides, we find that \(r = \sqrt[3]{343}\) feet.

Step 7 ：Simplifying, we find that \(r = 7\) feet.

Step 8 ：The surface area of a sphere is given by the formula \(S = 4 \pi r^2\).

Step 9 ：Substituting \(r = 7\) feet into the formula, we get \(S = 4 \pi (7)^2\) square feet.

Step 10 ：Simplifying, we find that \(S = 4 \pi \times 49\) square feet.

Step 11 ：Simplifying further, we find that \(S = 196 \pi\) square feet.

Step 12 ：So, the surface area of the sphere is \(\boxed{196 \pi}\) square feet.