A sphere and a cylinder have equal volume and equal radius. The ratio of the curved surface area of the cylinder to that of the sphere is

Options

A. `4:3`
B. `2:3`
C. `3:2`
D. `3:4`

Answer

To solve the problem, we need to find the ratio of the curved surface area of a cylinder to that of a sphere, given that they have equal volumes and equal radii.

### Step-by-Step Solution:

1. **Understand the formulas for volume**:
- The volume \( V \) of a cylinder is given by:
\[ V_{cylinder} = \pi r^2 h \]
- The volume \( V \) of a sphere is given by:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]

2. **Set the volumes equal**:
Since the cylinder and the sphere have equal volumes:
\[ \pi r^2 h = \frac{4}{3} \pi r^3 \]

3. **Cancel out common terms**:
We can cancel \( \pi \) from both sides:
\[ r^2 h = \frac{4}{3} r^3 \]

4. **Solve for height \( h \)**:
Divide both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[ h = \frac{4}{3} r \]

5. **Calculate the curved surface area (CSA)**:
- The curved surface area of the cylinder is:
\[ CSA_{cylinder} = 2 \pi r h \]
- Substitute \( h \) from the previous step:
\[ CSA_{cylinder} = 2 \pi r \left(\frac{4}{3} r\right) = \frac{8}{3} \pi r^2 \]

- The curved surface area of the sphere is:
\[ CSA_{sphere} = 4 \pi r^2 \]

6. **Find the ratio of the curved surface areas**:
Now, we find the ratio of the curved surface area of the cylinder to that of the sphere:
\[ \text{Ratio} = \frac{CSA_{cylinder}}{CSA_{sphere}} = \frac{\frac{8}{3} \pi r^2}{4 \pi r^2} \]

7. **Simplify the ratio**:
Cancel \( \pi r^2 \) from the numerator and denominator:
\[ \text{Ratio} = \frac{\frac{8}{3}}{4} = \frac{8}{3} \times \frac{1}{4} = \frac{8}{12} = \frac{2}{3} \]

### Final Answer:
The ratio of the curved surface area of the cylinder to that of the sphere is:
\[ \frac{2}{3} \]