The volume of a right circular cone is equal to the volume of that right circular cylinder whose height is 27 cm and diameter of its base is 30 cm. If the height of the cone is 25 cm, then what will be the diameter of its base?

Options

A. A) 24cm
B. B) 54cm
C. C) 27cm
D. D) 35cm

Answer

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

### Step 1: Understand the formulas for volume
The volume \( V \) of a right circular cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r_1^2 h_1 \]
where \( r_1 \) is the radius of the cone's base and \( h_1 \) is the height of the cone.

The volume \( V \) of a right circular cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r_2^2 h_2 \]
where \( r_2 \) is the radius of the cylinder's base and \( h_2 \) is the height of the cylinder.

### Step 2: Identify the given values
From the problem, we have:
- Height of the cylinder \( h_2 = 27 \) cm
- Diameter of the cylinder \( = 30 \) cm, thus the radius \( r_2 = \frac{30}{2} = 15 \) cm
- Height of the cone \( h_1 = 25 \) cm

### Step 3: Set the volumes equal
Since the volumes of the cone and cylinder are equal, we can set the two volume formulas equal to each other:
\[ \frac{1}{3} \pi r_1^2 h_1 = \pi r_2^2 h_2 \]

### Step 4: Cancel out \( \pi \) from both sides
We can simplify the equation by canceling \( \pi \) from both sides:
\[ \frac{1}{3} r_1^2 h_1 = r_2^2 h_2 \]

### Step 5: Substitute the known values
Substituting the known values into the equation:
\[ \frac{1}{3} r_1^2 (25) = (15)^2 (27) \]

### Step 6: Calculate the right side
Calculating \( (15)^2 \) and multiplying by \( 27 \):
\[ (15)^2 = 225 \]
\[ 225 \times 27 = 6075 \]
So, we have:
\[ \frac{25}{3} r_1^2 = 6075 \]

### Step 7: Solve for \( r_1^2 \)
To isolate \( r_1^2 \), multiply both sides by \( \frac{3}{25} \):
\[ r_1^2 = 6075 \times \frac{3}{25} \]
Calculating \( 6075 \times \frac{3}{25} \):
\[ 6075 \div 25 = 243 \]
\[ 243 \times 3 = 729 \]
Thus, we find:
\[ r_1^2 = 729 \]

### Step 8: Find \( r_1 \)
Taking the square root of both sides:
\[ r_1 = \sqrt{729} = 27 \text{ cm} \]

### Step 9: Calculate the diameter of the cone
The diameter \( d \) of the cone is given by:
\[ d = 2r_1 = 2 \times 27 = 54 \text{ cm} \]

### Final Answer
The diameter of the base of the cone is \( 54 \) cm.

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