A solid right circular cylinder and a solid sphere share the same radius. If the ratio of the volume of the cylinder to the volume of the sphere is 9 : 2, what is the ratio of the total surface area of the cylinder to the total surface area of the sphere?

The correct option is 7 : 2.

Let $r$ be the common radius shared by both the solid right circular cylinder and the solid sphere, and let $h$ be the height of the cylinder.

Step 1: Express the volume of each solid.

The volume of the cylinder ($V_{\text{cylinder}}$) is:

$V_{\text{cylinder}}=\pi r^{2}h$

The volume of the sphere ($V_{\text{sphere}}$) is:

$V_{\text{sphere}}=\frac{4}{3}\pi r^3$

Step 2: Determine the relationship between height and radius using the volume ratio.

We are given that the ratio of the volume of the cylinder to the volume of the sphere is 9 : 2:

$\frac{V_{\text{cylinder}}}{V_{\text{sphere}}}=\frac{9}{2}$

Substitute the volume formulas into this ratio:

$\frac{\pi r^{2}h}{\frac{4}{3}\pi r^3}=\frac{9}{2}$

Simplifying the left-hand side:

$\frac{3h}{4r}=\frac{9}{2}$

Solve for $h$:

$h=\frac{9}{2}\times \frac{4r}{3}=6r$

Step 3: Calculate the total surface areas and their ratio.

The total surface area of the cylinder ($A_{\text{cylinder}}$) includes the curved surface area plus the area of the two circular bases:

$A_{\text{cylinder}}=2\pi rh+2\pi r^2=2\pi r(h+r)$

Substituting $h=6r$:

$A_{\text{cylinder}}=2\pi r(6r+r)=2\pi r\left(7r\right)=14\pi r^2$

The total surface area of the sphere ($A_{\text{sphere}}$) is:

$A_{\text{sphere}}=4\pi r^2$

Now, find the ratio of their total surface areas:

$\frac{A_{\text{cylinder}}}{A_{\text{sphere}}}=\frac{14\pi r^2}{4\pi r^2}=\frac{7}{2}$

Hence, the ratio of the total surface area of the cylinder to the total surface area of the sphere is 7 : 2.