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

Answer: (D)

$\frac{\pi}{100\pi}cm^2$

Answer 2

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 $100\pi^2 cm$.