Metallic spheres of radii 6 cm, 8 cm and 10 cm respectively are melted and recasted into a single solid sphere. Taking `pi = 3.1,` find the surface area of the solid sphere formed.

Answer

To solve the problem of finding the surface area of the solid sphere formed by melting three metallic spheres of radii 6 cm, 8 cm, and 10 cm, we will follow these steps:

### Step 1: Calculate the volumes of the individual spheres.
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.

1. **Volume of the sphere with radius 6 cm:**
\[ V_1 = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \times 216 = 288 \pi \, \text{cm}^3 \]

2. **Volume of the sphere with radius 8 cm:**
\[ V_2 = \frac{4}{3} \pi (8)^3 = \frac{4}{3} \pi \times 512 = \frac{2048}{3} \pi \, \text{cm}^3 \]

3. **Volume of the sphere with radius 10 cm:**
\[ V_3 = \frac{4}{3} \pi (10)^3 = \frac{4}{3} \pi \times 1000 = \frac{4000}{3} \pi \, \text{cm}^3 \]

### Step 2: Add the volumes of the three spheres.
Now, we will add the volumes of the three spheres:
\[ V_{total} = V_1 + V_2 + V_3 = 288 \pi + \frac{2048}{3} \pi + \frac{4000}{3} \pi \]
To add these, we need a common denominator:
\[ V_{total} = 288 \pi + \left(\frac{2048 + 4000}{3}\right) \pi = 288 \pi + \frac{6048}{3} \pi \]
Convert \( 288 \) to a fraction with a denominator of 3:
\[ 288 = \frac{864}{3} \]
Now we can add:
\[ V_{total} = \left(\frac{864 + 6048}{3}\right) \pi = \frac{6912}{3} \pi \, \text{cm}^3 \]

### Step 3: Set the total volume equal to the volume of the new sphere.
Let the radius of the new sphere be \( r \). The volume of the new sphere is:
\[ V_{new} = \frac{4}{3} \pi r^3 \]
Setting the total volume equal to the volume of the new sphere:
\[ \frac{6912}{3} \pi = \frac{4}{3} \pi r^3 \]
We can cancel \( \frac{4}{3} \pi \) from both sides:
\[ 6912 = 4 r^3 \]
Dividing both sides by 4:
\[ r^3 = 1728 \]
Taking the cube root:
\[ r = \sqrt[3]{1728} = 12 \, \text{cm} \]

### Step 4: Calculate the surface area of the new sphere.
The formula for the surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
Substituting \( r = 12 \) cm:
\[ A = 4 \pi (12)^2 = 4 \pi \times 144 = 576 \pi \, \text{cm}^2 \]
Using \( \pi = 3.1 \):
\[ A = 576 \times 3.1 = 1785.6 \, \text{cm}^2 \]

### Final Answer:
The surface area of the solid sphere formed is \( 1785.6 \, \text{cm}^2 \).

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