Question 8
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.
\[
V=972 \pi \mathrm{cm}^{3} ; r=?
\]
$12 \mathrm{~cm}$
$3 \mathrm{~cm}$
$729 \mathrm{~cm}$
$9 \mathrm{~cm}$
Answer
\(\boxed{9 \, \mathrm{cm}}\) is the final answer.
Step 1 ：Given the volume of the sphere \(V = 972 \pi \, \mathrm{cm}^{3}\), we are asked to find the radius \(r\).
Step 2 ：The formula for the volume of a sphere is \(V = \frac{4}{3} \pi r^{3}\). We can rearrange this formula to solve for \(r\): \(r = \left(\frac{3V}{4\pi}\right)^{\frac{1}{3}}\).
Step 3 ：Substitute the given volume into the formula: \(r = \left(\frac{3 \times 972 \pi}{4\pi}\right)^{\frac{1}{3}}\).
Step 4 ：Simplify the expression to find the radius: \(r \approx 9 \, \mathrm{cm}\).
Step 5 ：\(\boxed{9 \, \mathrm{cm}}\) is the final answer.
