A solid cylinder of lead 8 m high and 2 m radius is melted and recast into a cone of radius 1.5 m. Find the height of the cone.

Options

A. 16.67 m
B. 21.35 m
C. 42.67 m
D. 31.35 m

Answer

To solve the problem, we need to find the height of a cone that is formed by melting a solid cylinder of lead. The volumes of the cylinder and the cone will be equal since the lead is conserved during the melting and recasting process.

### 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 the given cylinder:
- Radius \( r = 2 \, \text{m} \)
- Height \( h = 8 \, \text{m} \)

Substituting the values:
\[ V_{\text{cylinder}} = \pi (2)^2 (8) = \pi (4)(8) = 32\pi \, \text{m}^3 \]

2. **Calculate the Volume of the Cone**:
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

For the cone:
- Radius \( r = 1.5 \, \text{m} \)
- Height \( h \) is what we need to find.

Substituting the radius into the volume formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi (1.5)^2 h = \frac{1}{3} \pi (2.25) h = \frac{2.25\pi}{3} h \, \text{m}^3 \]

3. **Set the Volumes Equal**:
Since the volume of the cylinder is equal to the volume of the cone, we can set the equations equal to each other:
\[ 32\pi = \frac{2.25\pi}{3} h \]

4. **Cancel \(\pi\) from Both Sides**:
We can divide both sides by \(\pi\) (assuming \(\pi \neq 0\)):
\[ 32 = \frac{2.25}{3} h \]

5. **Multiply Both Sides by 3**:
To eliminate the fraction, multiply both sides by 3:
\[ 96 = 2.25h \]

6. **Solve for \( h \)**:
Now, divide both sides by 2.25 to find \( h \):
\[ h = \frac{96}{2.25} \]

To simplify \( \frac{96}{2.25} \):
\[ h = \frac{96 \times 100}{225} = \frac{9600}{225} \approx 42.67 \, \text{m} \]

### Final Answer:
The height of the cone is approximately \( 42.67 \, \text{m} \).