The volume of a right circular cylinder is equal to the volume of that right circular cone whose height is 108 cm and diameter of base is 30 cm. If the height of the cylinder is 9 cm, the diameter of its base is

Options

A. 30 cm
B. 60 cm
C. 50 cm
D. 40 cm

Answer

To solve the problem, we need to find the diameter of the base of a right circular cylinder, given that its volume is equal to the volume of a right circular cone. Here are the steps to find the solution:

### Step-by-Step Solution:

1. **Understand the volumes**:
The volume of a right circular cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height.

The volume of a right circular cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height.

2. **Given data for the cone**:
- Height \( h = 108 \) cm
- Diameter of the base \( d = 30 \) cm, thus the radius \( r = \frac{d}{2} = \frac{30}{2} = 15 \) cm.

3. **Calculate the volume of the cone**:
Substitute the values into the cone volume formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi (15)^2 (108) \]
\[ = \frac{1}{3} \pi (225) (108) \]
\[ = \frac{1}{3} \pi (24300) \]
\[ = 8100 \pi \text{ cm}^3 \]

4. **Set the volume of the cylinder equal to the volume of the cone**:
We know that the volume of the cylinder is equal to the volume of the cone:
\[ V_{\text{cylinder}} = 8100 \pi \]

5. **Use the height of the cylinder**:
The height of the cylinder is given as \( h = 9 \) cm. We can write:
\[ V_{\text{cylinder}} = \pi r^2 (9) \]
Setting this equal to the volume of the cone:
\[ \pi r^2 (9) = 8100 \pi \]

6. **Cancel \( \pi \) from both sides**:
\[ 9 r^2 = 8100 \]

7. **Solve for \( r^2 \)**:
Divide both sides by 9:
\[ r^2 = \frac{8100}{9} \]
\[ r^2 = 900 \]

8. **Find \( r \)**:
Taking the square root:
\[ r = \sqrt{900} = 30 \text{ cm} \]

9. **Calculate the diameter**:
The diameter \( d \) of the cylinder is:
\[ d = 2r = 2 \times 30 = 60 \text{ cm} \]

### Final Answer:
The diameter of the base of the cylinder is **60 cm**.