A hemispherical basin 150 cm in diameter holds water one hundred and twenty times as much a cylindrical tube. If the height of the tube is 15 cm , then the diameter of the tube ( in cm ) is :

Options

A. 23
B. 24
C. 25
D. 26

Answer

To solve the problem step by step, we need to find the diameter of a cylindrical tube given that a hemispherical basin holds water that is 120 times the volume of the cylindrical tube.

### Step-by-Step Solution:

1. **Find the Radius of the Hemispherical Basin:**
- The diameter of the hemispherical basin is given as 150 cm.
- The radius \( r \) of the hemispherical basin is half of the diameter:
\[ r = \frac{150}{2} = 75 \text{ cm} \]

**Hint:** Remember that the radius is always half of the diameter.

2. **Calculate the Volume of the Hemispherical Basin:**
- The formula for the volume \( V \) of a hemisphere is:
\[ V = \frac{2}{3} \pi r^3 \]
- Substituting the radius \( r = 75 \) cm:
\[ V = \frac{2}{3} \pi (75)^3 \]

3. **Calculate \( 75^3 \):**
- First, calculate \( 75^3 \):
\[ 75^3 = 75 \times 75 \times 75 = 421875 \]
- Now substitute back into the volume formula:
\[ V = \frac{2}{3} \pi (421875) \]

4. **Relate the Volume of the Hemispherical Basin to the Volume of the Cylindrical Tube:**
- According to the problem, the volume of the hemispherical basin is 120 times the volume of the cylindrical tube:
\[ V_{\text{hemisphere}} = 120 \times V_{\text{cylinder}} \]
- The volume \( V \) of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]
- Here, \( h = 15 \) cm is the height of the cylindrical tube.

5. **Set Up the Equation:**
- Substitute the volumes into the equation:
\[ \frac{2}{3} \pi (421875) = 120 \times \pi r^2 (15) \]
- Cancel \( \pi \) from both sides:
\[ \frac{2}{3} (421875) = 120 \times r^2 \times 15 \]

6. **Simplify the Equation:**
- Calculate \( 120 \times 15 = 1800 \):
\[ \frac{2}{3} (421875) = 1800 r^2 \]
- Calculate \( \frac{2}{3} \times 421875 = 281250 \):
\[ 281250 = 1800 r^2 \]

7. **Solve for \( r^2 \):**
- Divide both sides by 1800:
\[ r^2 = \frac{281250}{1800} \]
- Calculate \( \frac{281250}{1800} = 156.25 \):
\[ r^2 = 156.25 \]

8. **Find the Radius \( r \):**
- Take the square root of both sides:
\[ r = \sqrt{156.25} = 12.5 \text{ cm} \]

9. **Calculate the Diameter of the Cylindrical Tube:**
- The diameter \( D \) is twice the radius:
\[ D = 2r = 2 \times 12.5 = 25 \text{ cm} \]

### Final Answer:
The diameter of the cylindrical tube is **25 cm**.

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