If a hemispherical dome has an inner radius 21 m then its volume ( in `m^(3)`) is :

Options

A. `4910 m^(3) `
B. `18354 m^(3) `
C. `19404 m^(3)`
D. none of these

Answer

To find the volume of a hemispherical dome with an inner radius of 21 m, we can use the formula for the volume of a hemisphere. The formula is:

\[ V = \frac{2}{3} \pi r^3 \]

where \( V \) is the volume and \( r \) is the radius.

### Step-by-Step Solution:

1. **Identify the radius**:
The radius \( r \) of the hemispherical dome is given as 21 m.

2. **Substitute the radius into the volume formula**:
We substitute \( r = 21 \) m into the volume formula:
\[ V = \frac{2}{3} \pi (21)^3 \]

3. **Calculate \( 21^3 \)**:
First, we need to calculate \( 21^3 \):
\[ 21^3 = 21 \times 21 \times 21 = 441 \times 21 = 9261 \]

4. **Substitute \( 21^3 \) back into the volume formula**:
Now we can substitute \( 9261 \) back into the formula:
\[ V = \frac{2}{3} \pi (9261) \]

5. **Use the value of \( \pi \)**:
We will use \( \pi \approx \frac{22}{7} \) for calculations:
\[ V = \frac{2}{3} \times \frac{22}{7} \times 9261 \]

6. **Multiply the constants**:
First, calculate \( \frac{2 \times 22}{3 \times 7} \):
\[ \frac{44}{21} \]

7. **Now multiply by \( 9261 \)**:
\[ V = \frac{44}{21} \times 9261 \]

8. **Calculate \( \frac{44 \times 9261}{21} \)**:
First, calculate \( 44 \times 9261 = 407484 \).
Now divide by \( 21 \):
\[ V = \frac{407484}{21} = 19404 \]

9. **Final Volume**:
Therefore, the volume of the hemispherical dome is:
\[ V = 19404 \, m^3 \]

### Final Answer:
The volume of the hemispherical dome is **19404 m³**.