A metal sphere having a radius of 10 centimeters is melted down and molded into 8 identical smaller solid spheres. What is the ratio of the surface area of the original sphere to the total surface area of all 8 smaller spheres?

Correct option is A

​Given :

Radius of original sphere = 10 cm

It is melted to form 8 identical smaller spheres

Volume is conserved when melting.

Formula Used :

Volume of a sphere:
V =$$\frac{4}{3}\pi r^3$$​​

Surface area of a sphere:
S = $$4\pi r^2$$​​

Solution :

Volume of large sphere = Total volume of 8 small spheres
​$$\frac{4}{3}\pi (10)^3 = 8 \times \frac{4}{3}\pi r^3$$​​

1000 =$$8r^3$$​​
​$$r^3 = \frac{1000}{8} = 125$$​​
r = 5 cm

Surface areas

Original sphere:
​$$S_1 = 4\pi (10)^2 = 400\pi$$​​

Each small sphere:
​$$S_2 = 4\pi (5)^2 = 100\pi$$​​

Total surface area of 8 small spheres:
8 $$\times 100\pi = 800\pi$$​​

Ratio (Original SA : Total SA of 8 small spheres)

​$$= \frac{400\pi}{800\pi} = \frac{1}{2}$$​​