If ratio between volume of a cylinder and volume of sphere is 3 : 1, then find the ratio between total surface area of cylinder to total surface area of sphere, radius of sphere is equal to radius of cylinder?

The correct option is 5 : 2.

Let the radius of the sphere be $r$ and the radius of the cylinder also be $r$, since they are given to be equal. Let the height of the cylinder be $h$.

First, we express the volume of the cylinder ($V_c$) and the volume of the sphere ($V_s$):
$V_c=\pi r^{2}h$
$V_s=\frac{4}{3}\pi r^3$

We are given that the ratio of the volume of the cylinder to the volume of the sphere is 3 : 1.
$\frac{V_c}{V_s}=\frac{3}{1}$

Substitute the formulas for volume into the ratio:
$\frac{\pi r^{2}h}{\frac{4}{3}\pi r^3}=3$

Simplify the equation by canceling $\pi$ and $r^2$:
$\frac{h}{\frac{4}{3}r}=3$
$\frac{3h}{4r}=3$

Dividing both sides by 3, we get:
$\frac{h}{4r}=1$
$h=4r$

Now, we find the ratio between the total surface area of the cylinder ($T{S}_c$) and the total surface area of the sphere ($T{S}_s$).
The formula for the total surface area of a cylinder is:
$T{S}_c=2\pi r(r+h)$

Substitute $h=4r$ into the cylinder's surface area equation:
$T{S}_c=2\pi r(r+4r)=2\pi r\left(5r\right)=10\pi r^2$

The total surface area of the sphere is:
$T{S}_s=4\pi r^2$

Now, find the ratio of their total surface areas:
$\frac{T{S}_c}{T{S}_s}=\frac{10\pi r^2}{4\pi r^2}$

Cancel $\pi r^2$ from the numerator and the denominator:
$\frac{T{S}_c}{T{S}_s}=\frac{10}{4}=\frac{5}{2}$

Therefore, the ratio of the total surface area of the cylinder to the total surface area of the sphere is 5 : 2.