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

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**Solution:**

1. Let the radius of both the cylinder and sphere be *r*.

2. Volume of cylinder = $\pi r^2 h$

3. Volume of sphere = $\frac{4}{3} \pi r^3$

4. Given the ratio of volumes:
   $$\frac{\pi r^2 h}{\frac{4}{3} \pi r^3} = \frac{3}{1}$$

5. Simplifying:
   $$\frac{h}{\frac{4}{3} r} = 3 \implies h = 3 	imes \frac{4}{3} r = 4r$$

6. Total surface area (TSA) of cylinder = $2 \pi r (r + h) = 2 \pi r (r + 4r) = 2 \pi r (5r) = 10 \pi r^2$

7. TSA of sphere = $4 \pi r^2$

8. Required ratio:
   $$\frac{	ext{TSA of cylinder}}{	ext{TSA of sphere}} = \frac{10 \pi r^2}{4 \pi r^2} = \frac{10}{4} = \frac{5}{2}$$

**Answer:** $5 : 2$

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