A right circular cone of height 20 cm and base radius 15 cm is melted and cast into smaller cones of equal sizes of height 5 cm and base radius 1.5 cm. The number of cones cast are

Options

A. a) 300
B. b) 150
C. c) 400
D. d) 100

Answer

To solve the problem, we need to find the number of smaller cones that can be formed from the volume of a larger cone.

### Step-by-Step Solution:

1. **Calculate the Volume of the Larger Cone:**
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
For the larger cone:
- Height \( h = 20 \) cm
- Base radius \( r = 15 \) cm

Substituting the values:
\[ V_{large} = \frac{1}{3} \pi (15)^2 (20) \]
\[ V_{large} = \frac{1}{3} \pi (225) (20) \]
\[ V_{large} = \frac{1}{3} \pi (4500) \]
\[ V_{large} = 1500 \pi \, \text{cm}^3 \]

2. **Calculate the Volume of a Smaller Cone:**
For the smaller cone:
- Height \( h = 5 \) cm
- Base radius \( r = 1.5 \) cm

Using the same volume formula:
\[ V_{small} = \frac{1}{3} \pi (1.5)^2 (5) \]
\[ V_{small} = \frac{1}{3} \pi (2.25) (5) \]
\[ V_{small} = \frac{1}{3} \pi (11.25) \]
\[ V_{small} = 3.75 \pi \, \text{cm}^3 \]

3. **Determine the Number of Smaller Cones:**
To find the number of smaller cones \( n \) that can be made from the larger cone, we divide the volume of the larger cone by the volume of one smaller cone:
\[ n = \frac{V_{large}}{V_{small}} = \frac{1500 \pi}{3.75 \pi} \]
The \( \pi \) cancels out:
\[ n = \frac{1500}{3.75} \]
\[ n = 400 \]

### Conclusion:
The number of smaller cones that can be cast from the larger cone is **400**.

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