The correct option is (D): \(\frac{\pi}{100\pi}cm^2\)
Let's solve the problem step by step without using LaTeX:
### Step 1: Volume of the original sphere
\(\text{Volume of sphere} = \frac{4}{3} \times \pi \times \text{radius}^3\)
For the original sphere with a radius of 10 cm:
\(\text{Volume} = \frac{4}{3} \times \pi \times (10)^3 = \frac{4}{3} \times \pi \times 1000 = \frac{4000}{3} \pi \, \text{cubic cm}\)
Since the large sphere is moulded into 8 equal spherical balls, the volume of each small ball will be:
\(\text{Volume of each small ball} = \frac{1}{8} \times \frac{4000}{3} \pi = \frac{500}{3} \pi \, \text{cubic cm}\)
\(\text{Volume of small ball} = \frac{4}{3} \times \pi \times \text{small radius}^3\)
Equating the volume of the small ball to \(\frac{500}{3} \pi\) we get:
\(\frac{4}{3} \times \pi \times (\text{small radius})^3 = \frac{500}{3} \pi\)
\(\frac{4}{3} \times (\text{small radius})^3 = \frac{500}{3}\)
\((\text{small radius})^3 = \frac{500}{4} = 125\)
\(\text{small radius} = \sqrt[3]{125} = 5 \, \text{cm}\)
Step 4: Find the surface area of each small ball
The surface area of a sphere is given by the formula:
\(\text{Surface area} = 4 \times \pi \times \text{radius}^2\)
For each small ball with a radius of 5 cm:
\(\text{Surface area} = 4 \times \pi \times (5)^2 = 4 \times \pi \times 25 = 100 \pi^2 \, \text{cm}\)
So, the surface area of each ball is \(100\pi^2 cm\).