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: We can find the ratio of the radius from the ratio of volume using the equation $V = \dfrac{4}{3}\pi {r^3}$. Then we can find the required ratio by substituting the ratio of the radius in the equation of surface area. The equation of the surface area of a sphere is $S = 4\pi {r^2}$.

Complete step by step solution: Let the volume of the spheres be ${V_1}$and ${V_2}$ then their respective radius will be ${r_1}$ and ${r_2}$.
From the question, ${V_1} = 8{V_2}$
$ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = 8$
We know that the volume of a sphere is $V = \dfrac{4}{3}\pi {r^3}$
$ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{\dfrac{4}{3}\pi {r_1}^3}}{{\dfrac{4}{3}\pi {r_2}^3}}8$
Cancelling the common terms, we get,
$ \Rightarrow {\left( {\dfrac{{{r_1}}}{{{r_2}}}} \right)^3} = 8$
Taking the cube root, we get,
$ \Rightarrow \dfrac{{{r_1}}}{{{r_2}}} = 2$ … (1)
We know that surface area of a sphere is $S = 4\pi {r^2}$. Then the ratio of surface area is given by,
$\dfrac{{{S_1}}}{{{S_2}}} = \dfrac{{4\pi {r_1}^2}}{{4\pi {r_2}^2}}$
Cancelling the common terms, we get,
$\dfrac{{{S_1}}}{{{S_2}}} = {\left( {\dfrac{{{r_1}}}{{{r_2}}}} \right)^2}$
Substituting equation (1) we get
$\dfrac{{{S_1}}}{{{S_2}}} = {\left( 2 \right)^2} = 4$
$ \Rightarrow {S_1} = 4{S_2}$
Therefore, the ratio of the surface area of the spheres is $4:1$.

So, the correct answer is option B.

Note: A ratio is used to compare the quantity of something to another thing. We can multiply or divide a ratio with the same value. While handling ratios, their order is important. The volume of a body is the space occupied by the body in the three-dimensional space. For a sphere, it is given by $V = \dfrac{4}{3}\pi {r^3}$. The surface area of a body is the total area of all the surfaces of the object.
An alternate approach to this problem is by taking proportionality.
We have a volume of 2 spheres in the ratio 8:1
We know that volume of the sphere is proportional to the cube of the radius
$V\alpha {r^3}$ .. (1)
and surface area of the sphere is proportional to the square of the radius
$S\alpha {r^2}$
Taking square root. We get,
$ \Rightarrow {S^{\dfrac{1}{2}}}\alpha r$.. (2)
Substituting (2) in (1),
$V\alpha {\left( {{S^{\dfrac{1}{2}}}} \right)^3}$
$ \Rightarrow V\alpha {S^{\dfrac{3}{2}}}$
$S\alpha {V^{\dfrac{2}{3}}}$
Therefore, the ratio of surface area is ${8^{\dfrac{2}{3}}} = {2^2} = 4$
So the required ratio is $4:1$.