Question

The volume of a sphere is 8 times that of another sphere. What is the ratio of their surface areas?
(A) $8:1$
(B) $4:1$
(C) $2:1$
(D) $4:3$

Answer 1

Hint: Start with assuming the radii of spheres to be different variables. Then compare their volumes using the formula of volume of sphere i.e. $\dfrac{4}{3}\pi {r^3}$ as per the condition given in the question. Finally compare their surface areas using the formula of surface area of sphere i.e. $4\pi {r^2}$ to get the final ratio.

Complete step-by-step answer:
According to the question, the volume of a sphere is 8 times that of another sphere.
Let $R$ and $r$ be the radii of respective spheres. Then we have:
$ \Rightarrow {V_R} = 8{V_r}$
We know that the volume of the sphere is given as $\dfrac{4}{3}\pi {r^3}$, where $r$ is the radius. Using this formula, we’ll get:
$
   \Rightarrow \dfrac{4}{3}\pi {R^3} = 8 \times \dfrac{4}{3}\pi {r^3} \\
   \Rightarrow {R^3} = 8{r^3} \\
   \Rightarrow {\left( {\dfrac{R}{r}} \right)^3} = 8 \\
   \Rightarrow \dfrac{R}{r} = 2{\text{ }}.....{\text{(1)}} \\
$
Thus the ratio of radii of spheres is $2:1$.
Further, we know that the surface area of the sphere is given as $4\pi {r^2}$. So using this formula, the ratio of surface areas of spheres will be:
$ \Rightarrow $Ratio $ = \dfrac{{4\pi {R^2}}}{{4\pi {r^2}}} = {\left( {\dfrac{R}{r}} \right)^2}$
Putting the value of ratio of radii from equation, we’ll get:
$ \Rightarrow $Ratio $ = {\left( {\dfrac{2}{1}} \right)^2} = \dfrac{4}{1}$
Thus the ratio of surface areas of the given spheres is $4:1$.

(B) is the correct option.

Note: Volume of sphere depends on the cube of its radius while the surface area depends on the square of the radius. Thus if the volume of one sphere is 8 times that of another, its radius is cube root of 8 i.e. 2 times that of another sphere. Thus its surface area is square of radius i.e. 4 times of other spheres.