If the radius of a sphere is thrice to that of a hemisphere , then what will be the ratio of their respective volumes ?

Options

A. a) `27:1`
B. b) `9:1`
C. c) `54:1`
D. d) `18:1`

Answer

To solve the problem of finding the ratio of the volumes of a sphere and a hemisphere when the radius of the sphere is three times that of the hemisphere, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Radii**:
- Let the radius of the hemisphere be \( r_2 \).
- According to the problem, the radius of the sphere \( r_1 \) is three times that of the hemisphere. Therefore, we can write:
\[ r_1 = 3r_2 \]

2. **Volume of the Sphere**:
- The formula for the volume \( V_s \) of a sphere is given by:
\[ V_s = \frac{4}{3} \pi r_1^3 \]
- Substituting \( r_1 = 3r_2 \) into the volume formula:
\[ V_s = \frac{4}{3} \pi (3r_2)^3 \]
- Simplifying this:
\[ V_s = \frac{4}{3} \pi (27r_2^3) = 36 \pi r_2^3 \]

3. **Volume of the Hemisphere**:
- The formula for the volume \( V_h \) of a hemisphere is given by:
\[ V_h = \frac{2}{3} \pi r_2^3 \]

4. **Finding the Ratio of Volumes**:
- Now, we need to find the ratio of the volume of the sphere to the volume of the hemisphere:
\[ \text{Ratio} = \frac{V_s}{V_h} = \frac{36 \pi r_2^3}{\frac{2}{3} \pi r_2^3} \]
- The \( \pi r_2^3 \) terms cancel out:
\[ \text{Ratio} = \frac{36}{\frac{2}{3}} = 36 \times \frac{3}{2} = 54 \]

5. **Final Answer**:
- The ratio of the volumes of the sphere to the hemisphere is:
\[ \text{Ratio} = 54:1 \]