The total surface area of a solid right circular cylinder is twice that of a solid sphere. If they have the same radii, the ratio of the volume of the cylinder to that of the sphere is given by

Options

A. `9:4`
B. `2:1`
C. `3:1`
D. `4:9`

Answer

To solve the problem, we need to find the ratio of the volume of a solid right circular cylinder to that of a solid sphere, given that the total surface area of the cylinder is twice that of the sphere and they have the same radius.

### Step-by-Step Solution:

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

2. **Set Up the Equation**:
According to the problem, the total surface area of the cylinder is twice that of the sphere:
\[ 2\pi r(h + r) = 2 \times 4\pi r^2 \]

3. **Simplify the Equation**:
Dividing both sides by \(2\pi\):
\[ r(h + r) = 4r^2 \]
Dividing both sides by \(r\) (since \(r \neq 0\)):
\[ h + r = 4r \]
Rearranging gives:
\[ h = 4r - r = 3r \]

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

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

6. **Find the Ratio of the Volumes**:
The ratio of the volume of the cylinder to the volume of the sphere is:
\[ \text{Ratio} = \frac{V_{\text{cylinder}}}{V_{\text{sphere}}} = \frac{3\pi r^3}{\frac{4}{3}\pi r^3} \]
Simplifying this:
\[ \text{Ratio} = \frac{3}{\frac{4}{3}} = 3 \times \frac{3}{4} = \frac{9}{4} \]

### Final Answer:
The ratio of the volume of the cylinder to that of the sphere is \(\frac{9}{4}\).