The radii of a sphere and a right circular cylinder are equal and their curved surface areas are also equal. The ratio of their volumes is

Options

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

Answer

To solve the problem, we need to find the ratio of the volumes of a sphere and a right circular cylinder given that their radii are equal and their curved surface areas are equal.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The curved surface area (CSA) of a sphere is given by:
\[ \text{CSA}_{\text{sphere}} = 4\pi r^2 \]
- The curved surface area of a right circular cylinder is given by:
\[ \text{CSA}_{\text{cylinder}} = 2\pi rh \]

2. **Set the Curved Surface Areas Equal**:
Since the curved surface areas are equal, we can set the two equations equal to each other:
\[ 4\pi r^2 = 2\pi rh \]

3. **Cancel Common Terms**:
We can cancel \( \pi \) from both sides:
\[ 4r^2 = 2rh \]

4. **Solve for Height (h)**:
Rearranging the equation to find \( h \):
\[ 4r^2 = 2rh \implies h = \frac{4r^2}{2r} = 2r \]

5. **Volume of the Sphere**:
The volume \( V \) of a sphere is given by:
\[ V_{\text{sphere}} = \frac{4}{3}\pi r^3 \]

6. **Volume of the Cylinder**:
The volume \( V \) of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Substituting \( h = 2r \):
\[ V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3 \]

7. **Find the Ratio of the Volumes**:
Now, we can find the ratio of the volumes:
\[ \text{Ratio} = \frac{V_{\text{sphere}}}{V_{\text{cylinder}}} = \frac{\frac{4}{3}\pi r^3}{2\pi r^3} \]
Cancelling \( \pi r^3 \) from both the numerator and the denominator:
\[ \text{Ratio} = \frac{4/3}{2} = \frac{4}{3} \cdot \frac{1}{2} = \frac{4}{6} = \frac{2}{3} \]

8. **Final Answer**:
The ratio of the volumes of the sphere to the cylinder is:
\[ \frac{2}{3} \]