The volume of a right circular cone is equal to the volume of that right circular cylinder whose height is 48 cm and diameter of its base is 20 cm. If the height of the cone is 16 cm, then what will be the diameter of its base?

Options

A. 25 cm
B. 45 cm
C. 60 cm
D. 130 cm

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. Let's break down the solution step by step.

### Step 1: Identify the given values
- Height of the cylinder (h_cylinder) = 48 cm
- Diameter of the cylinder's base = 20 cm
- Height of the cone (h_cone) = 16 cm

### Step 2: Calculate the radius of the cylinder
The radius of the cylinder (r_cylinder) can be calculated from its diameter:
\[ r_{cylinder} = \frac{diameter}{2} = \frac{20 \, cm}{2} = 10 \, cm \]

### Step 3: Write the formula for the volumes
The volume of a right circular cone (V_cone) is given by:
\[ V_{cone} = \frac{1}{3} \pi r_{cone}^2 h_{cone} \]
The volume of a right circular cylinder (V_cylinder) is given by:
\[ V_{cylinder} = \pi r_{cylinder}^2 h_{cylinder} \]

### Step 4: Set the volumes equal to each other
Since the volumes are equal:
\[ \frac{1}{3} \pi r_{cone}^2 h_{cone} = \pi r_{cylinder}^2 h_{cylinder} \]

### Step 5: Substitute the known values into the equation
Substituting the known values into the equation:
\[ \frac{1}{3} \pi r_{cone}^2 (16) = \pi (10^2) (48) \]
We can cancel \(\pi\) from both sides:
\[ \frac{1}{3} r_{cone}^2 (16) = (10^2) (48) \]

### Step 6: Simplify the equation
Now simplify the equation:
\[ \frac{16}{3} r_{cone}^2 = 100 \times 48 \]
Calculating \(100 \times 48\):
\[ 100 \times 48 = 4800 \]
So we have:
\[ \frac{16}{3} r_{cone}^2 = 4800 \]

### Step 7: Solve for \(r_{cone}^2\)
Multiply both sides by 3:
\[ 16 r_{cone}^2 = 4800 \times 3 \]
Calculating \(4800 \times 3\):
\[ 4800 \times 3 = 14400 \]
So:
\[ 16 r_{cone}^2 = 14400 \]
Now divide both sides by 16:
\[ r_{cone}^2 = \frac{14400}{16} = 900 \]

### Step 8: Find \(r_{cone}\)
Taking the square root of both sides:
\[ r_{cone} = \sqrt{900} = 30 \, cm \]

### Step 9: Calculate the diameter of the cone
The diameter of the cone (d_cone) is:
\[ d_{cone} = 2 \times r_{cone} = 2 \times 30 = 60 \, cm \]

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

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