Some solid metallic right circular cones, each with radius of the base 3 cm and height 4 cm are melted to form a solid sphere of radius 6 cm. The number of right circular cones is

Options

A. A)12
B. B)24
C. C)48
D. D)6

Answer

To find the number of solid metallic right circular cones that can be melted to form a solid sphere, we need to compare the volumes of the cones and the sphere.

### Step 1: Calculate the volume of one cone.
The formula for the volume \( V \) of a right circular 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.

Given:
- Radius of the cone \( r = 3 \) cm,
- Height of the cone \( h = 4 \) cm.

Substituting the values:

\[ V = \frac{1}{3} \pi (3)^2 (4) = \frac{1}{3} \pi (9)(4) = \frac{36}{3} \pi = 12 \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of the sphere.
The formula for the volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi R^3 \]

Where:
- \( R \) is the radius of the sphere.

Given:
- Radius of the sphere \( R = 6 \) cm.

Substituting the value:

\[ V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = \frac{864}{3} \pi = 288 \pi \text{ cm}^3 \]

### Step 3: Set up the equation to find the number of cones.
Let \( n \) be the number of cones. The total volume of \( n \) cones must equal the volume of the sphere:

\[ n \times \text{Volume of one cone} = \text{Volume of the sphere} \]

Substituting the volumes we calculated:

\[ n \times 12 \pi = 288 \pi \]

### Step 4: Solve for \( n \).
Dividing both sides by \( \pi \):

\[ n \times 12 = 288 \]

Now, divide both sides by 12:

\[ n = \frac{288}{12} = 24 \]

### Conclusion
The number of right circular cones is \( n = 24 \).

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