A cylindrical piece of metal of radius 2 cm and height 6 is shaped into a cone of same radius. The height of the cone is---

Options

A. 18 cm
B. 14 cm
C. 12 cm
D. 8 cm

Answer

To find the height of the cone formed from a cylindrical piece of metal with a radius of 2 cm and a height of 6 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Volume of the Cylinder**:
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height. For our cylinder:
- Radius \( r = 2 \) cm
- Height \( h = 6 \) cm

Substituting the values:
\[ V = \pi (2)^2 (6) = \pi \cdot 4 \cdot 6 = 24\pi \text{ cm}^3 \]

2. **Set the Volume of the Cone Equal to the Volume of the Cylinder**:
The volume of the cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 H \]
where \( H \) is the height of the cone. Since the cone has the same radius as the cylinder, we have:
- Radius \( r = 2 \) cm

Therefore, the volume of the cone can be expressed as:
\[ V = \frac{1}{3} \pi (2)^2 H = \frac{1}{3} \pi \cdot 4 H = \frac{4}{3} \pi H \]

3. **Equate the Volumes**:
Since the volume of the cylinder is equal to the volume of the cone, we can set the two equations equal to each other:
\[ 24\pi = \frac{4}{3} \pi H \]

4. **Solve for \( H \)**:
To eliminate \( \pi \) from both sides, we divide both sides by \( \pi \):
\[ 24 = \frac{4}{3} H \]
Now, multiply both sides by \( 3 \) to get rid of the fraction:
\[ 72 = 4H \]
Finally, divide both sides by \( 4 \):
\[ H = \frac{72}{4} = 18 \text{ cm} \]

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

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