Digital SAT Math Practice Questions -Advanced : Area and volume

Ans: C

A sphere and a right circular cylinder both have radius \(r\). The height of the cylinder is 18. We need to find the value of \(r\) for which the volume of the sphere is twice the volume of the cylinder.

The volume \(V_{\text{sphere}}\) of a sphere is given by:
\[
V_{\text{sphere}} = \frac{4}{3}\pi r^3
\]

The volume \(V_{\text{cylinder}}\) of a right circular cylinder is given by:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
where \(h = 18\). So,
\[
V_{\text{cylinder}} = \pi r^2 \cdot 18 = 18\pi r^2
\]

We want the volume of the sphere to be twice the volume of the cylinder:
\[
\frac{4}{3}\pi r^3 = 2 \cdot 18\pi r^2
\]

Simplify and solve for \(r\):
\[
\frac{4}{3}\pi r^3 = 36\pi r^2
\]

Divide both sides by \(\pi\):
\[
\frac{4}{3} r^3 = 36 r^2
\]

Divide both sides by \(r^2\) (assuming \(r \neq 0\)):
\[
\frac{4}{3} r = 36
\]

Multiply both sides by \(\frac{3}{4}\):
\[
r = 36 \cdot \frac{3}{4}
\]
\[
r = 27
\]

So, the correct answer is:
\[
\boxed{27}
\]