A hemispherical bowl of internal diameter 54cm contains a liquid. The liquid is to be filled in cylinder bottles of radius 3cm and height 9cm, then number of bottles that can be filled?

Options

A. a. 160
B. b. 162
C. c. 155
D. d. 150

Answer

To solve the problem, we need to find out how many cylindrical bottles can be filled with the liquid from a hemispherical bowl.

### Step-by-Step Solution:

1. **Find the radius of the hemispherical bowl:**
- The internal diameter of the bowl is given as 54 cm.
- Therefore, the radius \( r \) of the hemispherical bowl is:
\[ r = \frac{\text{diameter}}{2} = \frac{54 \text{ cm}}{2} = 27 \text{ cm} \]

2. **Calculate the volume of the hemispherical bowl:**
- The formula for the volume \( V \) of a hemisphere is:
\[ V = \frac{2}{3} \pi r^3 \]
- Substituting the radius:
\[ V = \frac{2}{3} \pi (27)^3 \]
- Calculate \( 27^3 \):
\[ 27^3 = 19683 \]
- Now substitute this value into the volume formula:
\[ V = \frac{2}{3} \pi (19683) = \frac{39366}{3} \pi = 13122 \pi \text{ cm}^3 \]

3. **Calculate the volume of one cylindrical bottle:**
- The formula for the volume \( V \) of a cylinder is:
\[ V = \pi r^2 h \]
- Given the radius \( r = 3 \) cm and height \( h = 9 \) cm:
\[ V = \pi (3)^2 (9) = \pi (9)(9) = 81 \pi \text{ cm}^3 \]

4. **Determine the number of bottles that can be filled:**
- Let \( n \) be the number of bottles that can be filled. The total volume of liquid from the hemisphere is equal to the volume of \( n \) bottles:
\[ 13122 \pi = n \times 81 \pi \]
- Cancel \( \pi \) from both sides:
\[ 13122 = 81n \]
- Now solve for \( n \):
\[ n = \frac{13122}{81} \]
- Calculate \( n \):
\[ n = 162 \]

### Conclusion:
The number of cylindrical bottles that can be filled with the liquid from the hemispherical bowl is **162**.