A solid metallic cylinder of base radius 3 cm and height 5 cm is melted to make n solid cones of height 1 cm and base radius 1 mm. The value of n is

Options

A. 450
B. 1350
C. 4500
D. 13500

Explanation

To find the value of \( n \), which represents the number of solid cones that can be made from a solid metallic cylinder, we will follow these steps:

### Step 1: Calculate the Volume of the Cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
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:
\[ V = \pi (3)^2 (5) = \pi (9)(5) = 45\pi \text{ cm}^3 \]

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

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

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

### Step 3: Calculate the Number of Cones \( n \)
Since the volume of the cylinder is melted to form \( n \) cones, we can set up the equation:
\[ \text{Volume of the cylinder} = n \times \text{Volume of one cone} \]
Substituting the volumes we calculated:
\[ 45\pi = n \left(\frac{0.01\pi}{3}\right) \]

### Step 4: Solve for \( n \)
To isolate \( n \), we can divide both sides by \(\frac{0.01\pi}{3}\):
\[ n = \frac{45\pi}{\frac{0.01\pi}{3}} = 45 \times \frac{3}{0.01} = 45 \times 300 = 13500 \]

Thus, the value of \( n \) is:
\[ \boxed{13500} \]