Question

If a solid sphere of radius 10 cm is moulded into 8 spherical solid balls of equal radius then the surface area of each ball is

• A. 60 \(\pi\) cm²
• B. \(\frac{\pi}{50\pi}\) cm²
• C. 75 \(\pi\) cm²
• D. \(\frac{\pi}{100\pi}cm^2\)

Answer 1

The Correct Option is D

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 \(100\pi^2  cm\).