A hemispherical roof is on a circular room. Inner diameter of the roof is equal to the greatest height of the roof. If `48510m^(3)` air is inside the room, find the height of the roof.

Options

A. 21 m
B. 18 m
C. 42 m
D. `16(1)/(2)m`

Answer

To solve the problem, we need to find the height of a hemispherical roof on a circular room given that the inner diameter of the roof is equal to the greatest height of the roof, and the volume of air inside the room is 48510 m³.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let the inner diameter of the hemispherical roof be \( D \).
- Since the greatest height of the roof is also \( D \), we can denote the radius of the hemispherical roof as \( R = \frac{D}{2} \).

2. **Height of the Cylindrical Part**:
- The height of the cylindrical part of the room can be expressed as:
\[ H = D - R = D - \frac{D}{2} = \frac{D}{2} = R \]
- Thus, the height of the cylindrical part is equal to the radius of the hemispherical part.

3. **Volume of the Room**:
- The total volume \( V \) of the room consists of the volume of the cylindrical part and the volume of the hemispherical part.
- The volume of the cylindrical part is given by:
\[ V_{cylinder} = \pi R^2 H \]
- The volume of the hemispherical part is given by:
\[ V_{hemisphere} = \frac{2}{3} \pi R^3 \]
- Therefore, the total volume can be expressed as:
\[ V = V_{cylinder} + V_{hemisphere} = \pi R^2 R + \frac{2}{3} \pi R^3 = \pi R^3 + \frac{2}{3} \pi R^3 = \frac{5}{3} \pi R^3 \]

4. **Setting Up the Equation**:
- We know the total volume of the room is 48510 m³:
\[ \frac{5}{3} \pi R^3 = 48510 \]

5. **Solving for \( R^3 \)**:
- Rearranging the equation gives:
\[ R^3 = \frac{48510 \times 3}{5 \pi} \]
- Calculating the right side:
\[ R^3 = \frac{145530}{5 \pi} \approx \frac{145530}{15.70796} \approx 9261 \]

6. **Finding \( R \)**:
- Now, take the cube root of both sides:
\[ R = \sqrt[3]{9261} = 21 \text{ m} \]

7. **Finding the Height**:
- Since the height \( H \) of the roof is equal to the diameter \( D \), we have:
\[ D = 2R = 2 \times 21 = 42 \text{ m} \]

### Final Answer:
The height of the roof is **42 meters**.