A solid metallic sphere of radius x cm is melted and then drawn into 126 cones each of radius 3.5 cm and height 3 cm. There is no wastage of material in this process. What is the value of x?

Options

A. 10.5
B. 7
C. 3.5
D. 21

Answer

To solve the problem, we need to find the radius \( x \) of a solid metallic sphere that is melted down to create 126 cones, each with a radius of 3.5 cm and a height of 3 cm.

### Step-by-Step Solution:

1. **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. For our cones:
- Radius \( r = 3.5 \) cm
- Height \( h = 3 \) cm

Plugging in the values:
\[ V = \frac{1}{3} \pi (3.5)^2 (3) \]
\[ V = \frac{1}{3} \pi (12.25) (3) = \frac{1}{3} \pi (36.75) = 12.25 \pi \text{ cm}^3 \]

2. **Calculate the Total Volume of 126 Cones**:
Since there are 126 cones, the total volume \( V_{total} \) is:
\[ V_{total} = 126 \times 12.25 \pi = 1545 \pi \text{ cm}^3 \]

3. **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. For our sphere:
\[ V = \frac{4}{3} \pi x^3 \]

4. **Set the Volume of the Sphere Equal to the Total Volume of the Cones**:
Since the volume of the sphere is equal to the total volume of the cones:
\[ \frac{4}{3} \pi x^3 = 1545 \pi \]

5. **Cancel \( \pi \) from Both Sides**:
\[ \frac{4}{3} x^3 = 1545 \]

6. **Multiply Both Sides by \( \frac{3}{4} \)**:
\[ x^3 = 1545 \times \frac{3}{4} = \frac{4635}{4} \]

7. **Calculate \( x^3 \)**:
\[ x^3 = 1158.75 \]

8. **Take the Cube Root**:
To find \( x \), take the cube root of both sides:
\[ x = \sqrt[3]{1158.75} \]
Approximating \( x \):
\[ x \approx 10.5 \text{ cm} \]

### Final Answer:
The value of \( x \) is \( 10.5 \) cm.