A metallic sphere of radius 10.5 cm is melted and then recast into small cones each radius 3.5 cm and height 3 cm. The number of cones thus formed is

Options

A. 140
B. 132
C. 112
D. 126

Answer

To solve the problem, we need to find out how many small cones can be formed from a melted metallic sphere. We will do this by calculating the volumes of both the sphere and the cones, and then using these volumes to find the number of cones.

### Step 1: 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. Here, the radius of the sphere is \( 10.5 \) cm.

Substituting the value of \( r \):

\[ V = \frac{4}{3} \times \frac{22}{7} \times (10.5)^3 \]

Calculating \( (10.5)^3 \):

\[ 10.5^3 = 1157.625 \]

Now substituting this back into the volume formula:

\[ V = \frac{4}{3} \times \frac{22}{7} \times 1157.625 \]

Calculating the volume:

\[ V \approx 4851 \text{ cm}^3 \]

### Step 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 of the cone. Here, the radius of the cone is \( 3.5 \) cm and the height is \( 3 \) cm.

Substituting the values:

\[ V = \frac{1}{3} \times \frac{22}{7} \times (3.5)^2 \times 3 \]

Calculating \( (3.5)^2 \):

\[ (3.5)^2 = 12.25 \]

Now substituting this back into the volume formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 12.25 \times 3 \]

The \( 3 \) in the numerator and denominator cancels out:

\[ V = \frac{22}{7} \times 12.25 \]

Calculating the volume:

\[ V \approx 38.5 \text{ cm}^3 \]

### Step 3: Calculate the Number of Cones
Let \( n \) be the number of cones formed. The total volume of the cones must equal the volume of the sphere:

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

Substituting the volumes we calculated:

\[ n \times 38.5 = 4851 \]

Now, solving for \( n \):

\[ n = \frac{4851}{38.5} \]

Calculating \( n \):

\[ n \approx 126 \]

Thus, the number of cones formed is **126**.

### Summary of Steps:
1. Calculate the volume of the sphere using the formula for the volume of a sphere.
2. Calculate the volume of one cone using the formula for the volume of a cone.
3. Set the total volume of the cones equal to the volume of the sphere and solve for the number of cones.