The radius of a cone with height 3 cm, whose volume is equal to the volume of a cylinder with radius 6 cm and height 9 cmn is

Options

A. 17 cm
B. 17.5 cm
C. 18 cm
D. 19 cm

Answer

To find the radius of the cone whose volume is equal to the volume of a cylinder, we can follow these steps:

### Step 1: Write the formulas for the volumes
The volume \( V \) of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the cone and \( h \) is its height.

The volume \( V \) of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi R^2 H \]
where \( R \) is the radius of the cylinder and \( H \) is its height.

### Step 2: Substitute the known values
From the problem, we know:
- The height of the cone \( h = 3 \) cm.
- The radius of the cylinder \( R = 6 \) cm.
- The height of the cylinder \( H = 9 \) cm.

Now, substituting these values into the volume formulas:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (3) = \pi r^2 \]
\[ V_{\text{cylinder}} = \pi (6^2)(9) = \pi (36)(9) = 324\pi \]

### Step 3: Set the volumes equal to each other
Since the volumes are equal:
\[ \pi r^2 = 324\pi \]

### Step 4: Cancel out \( \pi \)
We can divide both sides of the equation by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ r^2 = 324 \]

### Step 5: Solve for \( r \)
To find \( r \), take the square root of both sides:
\[ r = \sqrt{324} = 18 \text{ cm} \]

### Conclusion
The radius of the cone is \( 18 \) cm.

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