Three solid spheres of diameters 6 cm, 8 cm and 10 cm are melted to form a single solid sphere. The diameter of the new sphere is

Options

A. 6cm
B. 4.5 cm
C. 3cm
D. 12cm

Answer

To find the diameter of the new solid sphere formed by melting three solid spheres with diameters of 6 cm, 8 cm, and 10 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Find the Radii of the Spheres:**
- The radius of a sphere is half of its diameter.
- For the first sphere (diameter = 6 cm):
\[ R_1 = \frac{6}{2} = 3 \text{ cm} \]
- For the second sphere (diameter = 8 cm):
\[ R_2 = \frac{8}{2} = 4 \text{ cm} \]
- For the third sphere (diameter = 10 cm):
\[ R_3 = \frac{10}{2} = 5 \text{ cm} \]

2. **Calculate the Volumes of the Spheres:**
- The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
- Volume of the first sphere:
\[ V_1 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \text{ cm}^3 \]
- Volume of the second sphere:
\[ V_2 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cm}^3 \]
- Volume of the third sphere:
\[ V_3 = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \text{ cm}^3 \]

3. **Sum the Volumes:**
- The total volume \( V_{total} \) of the new sphere is:
\[ V_{total} = V_1 + V_2 + V_3 = 36 \pi + \frac{256}{3} \pi + \frac{500}{3} \pi \]
- Convert \( 36 \pi \) to a fraction with a denominator of 3:
\[ 36 \pi = \frac{108}{3} \pi \]
- Now add the volumes:
\[ V_{total} = \frac{108}{3} \pi + \frac{256}{3} \pi + \frac{500}{3} \pi = \frac{864}{3} \pi \text{ cm}^3 \]

4. **Set the Total Volume Equal to the Volume of the New Sphere:**
- Let \( R \) be the radius of the new sphere. The volume of the new sphere is:
\[ V_{new} = \frac{4}{3} \pi R^3 \]
- Set the total volume equal to the new sphere's volume:
\[ \frac{4}{3} \pi R^3 = \frac{864}{3} \pi \]
- Cancel \( \frac{4}{3} \pi \) from both sides:
\[ R^3 = 216 \]

5. **Calculate the Radius of the New Sphere:**
- Take the cube root of both sides:
\[ R = \sqrt[3]{216} = 6 \text{ cm} \]

6. **Find the Diameter of the New Sphere:**
- The diameter \( D \) is twice the radius:
\[ D = 2R = 2 \times 6 = 12 \text{ cm} \]

### Final Answer:
The diameter of the new solid sphere is **12 cm**.

---