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

Options

A. 1350
B. 135000
C. 45
D. None of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the volume of the metallic cylinder.
The formula for the volume of a cylinder is given by:

\[ V_{\text{cylinder}} = \pi r^2 h \]

Where:
- \( r \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylinder.

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

Substituting the values:

\[ V_{\text{cylinder}} = \pi (3)^2 (5) = \pi (9)(5) = 45\pi \, \text{cm}^3 \]

### Step 2: Calculate the volume of one solid cone.
The formula for the volume of a cone is given by:

\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]

Where:
- \( r \) is the radius of the base of the cone,
- \( h \) is the height of the cone.

Given:
- Radius of the cone, \( r = 1 \) mm = \( \frac{1}{10} \) cm (since 1 cm = 10 mm),
- Height of the cone, \( h = 1 \) mm = \( \frac{1}{10} \) cm.

Substituting the values:

\[ V_{\text{cone}} = \frac{1}{3} \pi \left(\frac{1}{10}\right)^2 \left(\frac{1}{10}\right) = \frac{1}{3} \pi \left(\frac{1}{100}\right) \left(\frac{1}{10}\right) = \frac{1}{3} \pi \left(\frac{1}{1000}\right) = \frac{\pi}{3000} \, \text{cm}^3 \]

### Step 3: Set up the equation to find \( n \).
Since the volume of the cylinder is melted to form \( n \) cones, we have:

\[ V_{\text{cylinder}} = n \times V_{\text{cone}} \]

Substituting the volumes we calculated:

\[ 45\pi = n \times \frac{\pi}{3000} \]

### Step 4: Solve for \( n \).
We can cancel \( \pi \) from both sides:

\[ 45 = n \times \frac{1}{3000} \]

Now, multiply both sides by 3000 to isolate \( n \):

\[ n = 45 \times 3000 \]

Calculating \( n \):

\[ n = 135000 \]

### Final Answer:
The value of \( n \) is \( 135000 \).

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