A right circular cone has a volume of `24pi` cubic inches. If the height of the cone is 2 inches, what is the radius, in inches, of the base of the cone?

Options

A. `2sqrt3`
B. 6
C. 12
D. 36

Explanation

To find the radius of the base of the cone, we can use the formula for the volume of a cone:

\[ V = \frac{1}{3} \pi r^2 h \]

where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.

### Step 1: Substitute the known values into the volume formula.

We are given:
- Volume \( V = 24\pi \) cubic inches
- Height \( h = 2 \) inches

Substituting these values into the formula, we have:

\[ 24\pi = \frac{1}{3} \pi r^2 (2) \]

### Step 2: Simplify the equation.

First, we can simplify the right side:

\[ 24\pi = \frac{2}{3} \pi r^2 \]

Next, we can cancel \( \pi \) from both sides (since \( \pi \) is not zero):

\[ 24 = \frac{2}{3} r^2 \]

### Step 3: Eliminate the fraction.

To eliminate the fraction, multiply both sides by 3:

\[ 3 \times 24 = 2r^2 \]

This simplifies to:

\[ 72 = 2r^2 \]

### Step 4: Solve for \( r^2 \).

Next, divide both sides by 2:

\[ r^2 = \frac{72}{2} \]

This simplifies to:

\[ r^2 = 36 \]

### Step 5: Find the radius \( r \).

Now, take the square root of both sides:

\[ r = \sqrt{36} \]

Thus, we find:

\[ r = 6 \]

### Conclusion

The radius of the base of the cone is \( 6 \) inches.

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