A hemispherical roof is on a circular room. Inner diameter of the roof is equal to the greatest height of the roof. If `48510 m^(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 the hemispherical roof given that the inner diameter of the roof is equal to its greatest height and that the volume of air inside the room is \(48510 \, m^3\).

### Step-by-Step Solution:

1. **Understand the Geometry**:
- The roof is hemispherical, and it sits on a circular room (cylinder).
- Let the inner diameter of the roof be \(d\). According to the problem, the greatest height of the roof (which is also the radius of the hemisphere) is equal to the diameter, so:
\[ H = d \]
- The radius \(R\) of the hemisphere is:
\[ R = \frac{d}{2} = \frac{H}{2} \]

2. **Volume of the Room**:
- The volume of the room consists of the volume of the cylinder plus the volume of the hemisphere.
- The volume \(V\) of the cylinder is given by:
\[ V_{cylinder} = \pi r^2 h \]
- The volume \(V\) of the hemisphere is given by:
\[ V_{hemisphere} = \frac{2}{3} \pi R^3 \]

3. **Relate the Volumes**:
- The total volume of the room can be expressed as:
\[ V_{total} = V_{cylinder} + V_{hemisphere} \]
- Substituting the expressions for the volumes:
\[ V_{total} = \pi r^2 H + \frac{2}{3} \pi R^3 \]

4. **Substituting for Radius**:
- Since \(R = \frac{H}{2}\), we can express \(r\) (the radius of the circular room) in terms of \(H\):
\[ r = \frac{H}{2} \]
- Substitute \(r\) into the volume of the cylinder:
\[ V_{cylinder} = \pi \left(\frac{H}{2}\right)^2 H = \pi \frac{H^2}{4} H = \frac{\pi H^3}{4} \]

5. **Substituting for Hemisphere Volume**:
- Now substitute \(R\) into the hemisphere volume:
\[ V_{hemisphere} = \frac{2}{3} \pi \left(\frac{H}{2}\right)^3 = \frac{2}{3} \pi \frac{H^3}{8} = \frac{\pi H^3}{12} \]

6. **Total Volume Equation**:
- Now, we can write the total volume as:
\[ V_{total} = \frac{\pi H^3}{4} + \frac{\pi H^3}{12} \]
- To combine these, find a common denominator (which is 12):
\[ V_{total} = \frac{3\pi H^3}{12} + \frac{\pi H^3}{12} = \frac{4\pi H^3}{12} = \frac{\pi H^3}{3} \]

7. **Set the Total Volume Equal to Given Volume**:
- We know the total volume is \(48510 \, m^3\):
\[ \frac{\pi H^3}{3} = 48510 \]
- Multiply both sides by 3:
\[ \pi H^3 = 145530 \]
- Divide by \(\pi\):
\[ H^3 = \frac{145530}{\pi} \]

8. **Calculate H**:
- Using \(\pi \approx 3.14\):
\[ H^3 \approx \frac{145530}{3.14} \approx 46300 \]
- Taking the cube root:
\[ H \approx \sqrt[3]{46300} \approx 36.5 \, m \]

### Final Answer:
The height of the roof \(H\) is approximately \(36.5 \, m\).