A solid metallic sphere of radius 10 cm is melted and recast into 125 identical spheres. What is the ratio of the surface area of the original sphere to the total surface area of 6 smaller spheres so formed?​

Correct option is B

Given:

Radius of original sphere R = 10 cm

Melted into 125 identical smaller spheres

radius of each small sphere r

Need ratio: (Surface area of original sphere) : (Total surface area of 6 small spheres)

Formula Used:

Volume =$$\frac{4}{3}\pi R^3$$​ 

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

Solution:

Volume =$$\frac{4}{3}\pi R^3$$ =125$$\times \frac{4}{3}\pi r^3$$​ 

​$$r = \frac{R}{5}$$​​

From volume conservation: r $$= \frac{10}{5}$$​= 2 cm.

Surface area of original sphere: S$$_O = 4\pi(10)^2 = 400\pi$$​​

Surface area of one small sphere: $$S_s = 4\pi(2)^2 = 16\pi$$​​

Total surface area of 6 small spheres: 6$$\times 16\pi = 96\pi$$​​

Required ratio $$= 400\pi : 96\pi = 400:96 = 2$$​5:6