Volumes of two spheres are in the ratio 64 : 27. The ratio of their surface areas is ______.

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KollegeApply · Accepted Answer

Volumes of two spheres are in the ratio 64 : 27. The ratio of their surface areas is 16 : 9. Explanation: Let the radii of the two spheres are r1 and r2, respectively. $\therefore$ Volume of the sphere of radius, r1 = V1 = $\frac{4}{3}$ pi"r"_1^3 ...(i) [∵ Volume of sphere = $\frac{4}{3}$pi (radius)3] And volume of the sphere of radius, r2 = V2 = $\frac{4}{3}$ pi"r"_2^3 ...(ii) Given, ratio of volumes = V1 : V2 = 64 : 27 $\Rightarrow$ `($\frac{4}{3}$ pi"r"_1^3)/($\frac{4}{3}$ pi"r"_2^3) = $\frac{64}{27}$` ...[Using equations (i) and (ii)] $\Rightarrow$ ("r"_1^3)/("r"_2^3) = $\frac{64}{27}$ $\Rightarrow$ "r"_1/"r"_2 = $\frac{4}{3}$ ...(iii) Now, ratio of surface area = (4 pi"r"_1^2)/(4 pi"r"_2^2) ...[∵ Surface area of a sphere = 4$\pi$ (radius)2] = "r"_1^2/"r"_2^2 = ("r"_1/"r"_2)^2 = ($\frac{4}{3}$)^2 ...[Using equation (iii)] = 16 : 9 Hence, the required ratio of their surface area is 16 : 9.

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