The radius of a metallic cylinder is 3 cm and its height is 5 cm. It is melted and moulded into small cones, each of height into small cones, each of height 1 cm and base radius 1 mm. The number of such cones formed, is

Options

A. a) 450
B. b) 1350
C. c) 8500
D. d) 13500

Answer

To solve the problem, we need to find the number of small cones that can be formed from a metallic cylinder when it is melted. We will use the formula for the volume of a cylinder and the volume of a cone.

### 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.

Given:
- Radius of the cylinder \( r = 3 \) cm
- Height of the cylinder \( h = 5 \) cm

Substituting the values:
\[ V = \pi (3)^2 (5) = \pi (9)(5) = 45\pi \text{ cm}^3 \]

2. **Calculate the Volume of One 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.

Given:
- Radius of the cone \( r = 1 \) mm = \( 0.1 \) cm (since 1 cm = 10 mm)
- Height of the cone \( h = 1 \) cm

Substituting the values:
\[ V = \frac{1}{3} \pi (0.1)^2 (1) = \frac{1}{3} \pi (0.01) = \frac{1}{300} \pi \text{ cm}^3 \]

3. **Set the Volume of the Cylinder Equal to the Total Volume of the Cones:**
Let \( n \) be the number of cones formed. The total volume of \( n \) cones is:
\[ n \times \frac{1}{300} \pi \]
Setting the volume of the cylinder equal to the total volume of the cones:
\[ 45\pi = n \times \frac{1}{300} \pi \]

4. **Cancel \( \pi \) from Both Sides:**
\[ 45 = n \times \frac{1}{300} \]

5. **Solve for \( n \):**
Multiply both sides by 300:
\[ n = 45 \times 300 = 13500 \]

Thus, the number of small cones formed is \( \boxed{13500} \).