The radius of the base and height of a metallic solid cylinder are 5 cm and 6 cm respectively. It is melted and recast into a solid cone of the same radius of base. The height of the cone is :

Options

A. 54 cm
B. 27 cm
C. 18 cm
D. 9 cm

Answer

To find the height of the cone formed by melting a metallic solid cylinder, we will follow these steps:

### Step 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 of the base,
- \( h \) is the height.

Given:
- Radius \( r = 5 \) cm,
- Height \( h = 6 \) cm.

Substituting the values:
\[ V = \pi (5)^2 (6) = \pi \times 25 \times 6 = 150\pi \text{ cm}^3 \]

### Step 2: Set up the volume of the cone
The volume \( V' \) of a cone is given by:
\[ V' = \frac{1}{3} \pi r'^2 h' \]
where:
- \( r' \) is the radius of the base of the cone,
- \( h' \) is the height of the cone.

Since the radius of the cone is the same as that of the cylinder, we have \( r' = 5 \) cm.

### Step 3: Equate the volumes
Since the cylinder is melted and recast into a cone, the volumes are equal:
\[ 150\pi = \frac{1}{3} \pi (5)^2 h' \]

### Step 4: Simplify the equation
Cancelling \( \pi \) from both sides:
\[ 150 = \frac{1}{3} (25) h' \]

### Step 5: Solve for \( h' \)
Multiplying both sides by 3:
\[ 450 = 25 h' \]
Now, divide both sides by 25:
\[ h' = \frac{450}{25} = 18 \text{ cm} \]

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

---