What is the least number of solid metallic spheres, each of 6 cm diameter, that should be melted and recast to form a solid metal cone whose height is 45 cm and diameter 12 cm ?

Answer

To solve the problem, we need to find the least number of solid metallic spheres, each with a diameter of 6 cm, that can be melted and recast into a solid metal cone with a height of 45 cm and a diameter of 12 cm.

### Step-by-Step Solution:

1. **Calculate the radius of the sphere**:
- The diameter of the sphere is 6 cm, so the radius (r) is:
\[ r = \frac{6}{2} = 3 \text{ cm} \]

2. **Calculate the volume of one sphere**:
- The formula for the volume (V) of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
- Substituting the radius:
\[ V = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \text{ cm}^3 \]

3. **Calculate the radius of the cone**:
- The diameter of the cone is 12 cm, so the radius (R) is:
\[ R = \frac{12}{2} = 6 \text{ cm} \]

4. **Calculate the volume of the cone**:
- The formula for the volume (V) of a cone is:
\[ V = \frac{1}{3} \pi R^2 h \]
- Substituting the radius and height:
\[ V = \frac{1}{3} \pi (6)^2 (45) = \frac{1}{3} \pi (36)(45) = \frac{1}{3} \pi (1620) = 540 \pi \text{ cm}^3 \]

5. **Set up the equation for the number of spheres**:
- Let \( n \) be the number of spheres. The total volume of \( n \) spheres must equal the volume of the cone:
\[ n \times 36 \pi = 540 \pi \]

6. **Solve for \( n \)**:
- Dividing both sides by \( \pi \):
\[ n \times 36 = 540 \]
- Now, divide both sides by 36:
\[ n = \frac{540}{36} = 15 \]

### Final Answer:
The least number of solid metallic spheres required is **15**.