The volumes of the two spheres are in the ratio $64 : 27$ . Find the ratio of their surface areas.

16 : 9

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9 : 16

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4 : 3

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3 : 4

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Solution

The correct option is D $16 : 9$
Let the radii of the two spheres be $r_{1}$ and $r_{2}$ .
Their volumes will be $V_{1}$ and $V_{2}$
$V_{1} = \frac{4}{3} π {r_{1}}^{3} V_{2} = \frac{4}{3} π {r_{2}}^{3}$
Given, $V_{1} : V_{2} = 64 : 27$
$\Rightarrow {r_{1}}^{3} : {r_{2}}^{3} = 64 : 27$
$\Rightarrow r_{1} : r_{2} = 4 : 3$

Their surface areas will be $s_{1}$ and $s_{2}$
$\Rightarrow s_{1} : s_{2} = 4 π {r_{1}}^{2} : 4 π {r_{2}}^{2} \Rightarrow s_{1} : s_{2} = {r_{1}}^{2} : {r_{2}}^{2} = 16 : 9$

Hence, option A is correct.

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