​What is the ratio of the surface areas of two spheres, if their volumes are in the ratio 8 : 27?​

Correct option is B

​Given:

The volumes of two spheres are in the ratio 8:27.

Formula Used:

Volume of a sphere:

V = $$\frac{4}{3} \pi$$ r³​

Surface area of a sphere:

A = 4$$\pi$$r²

Solution:

The volume of a sphere is proportional to the cube of its radius (V ∝ r³).

The surface area of a sphere is proportional to the square of its radius (A ∝ r²).

The radius of the first sphere as 'r₁'

The radius of the second sphere as 'r₂'

$$\frac{\text{Volume of the first sphere }}{\text{ Volume of the second sphere}}$$​ = 8 : 27

Therefore, $$\frac{(r_1)^3}{(r_2)^3} = \frac{8}{27}$$​​

Taking the cube root of both sides

$$\frac{(r_1)}{(r_2)}= \frac {2}{3}$$​​

$$\frac{\text{Surface Area of the first sphere }}{\text{Surface Area of the second sphere }}$$​ = $$\frac{(r_1)^2}{(r_2)^2}$$​​

$$\frac{(r_1)^2}{(r_2)^2} = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$$​

Therefore, the ratio of the surface areas of the two spheres is 4 : 9.

Thus, option (b) is right answer.