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The surface area of two spheres are in the ratio of $9 : 25$ . Then their volumes are in the ratio

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81 : 625

729 : 625

27 : 75

27 : 125

answer is D.

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Detailed Solution

Given, the ratio of surface area of two sphere = 9: 25.
The formula for the surface area of sphere is given by,
Surface area of sphere

= 4 π r^{2},

where r is its radius.
Let

r_{1}

and

r_{2}

be the respective radii of the two spheres.
So,

\frac{4 π r_{1}^{2}}{4 π r_{2}^{2}} = \frac{9}{25}

\Rightarrow \frac{r_{1}^{2}}{r_{2}^{2}} = \frac{9}{25}

\Rightarrow \frac{r_{1}}{r_{2}} = \frac{3}{5}

The formula for the volume of sphere is given by,
Volume of sphere, V

= \frac{4}{3} π r^{3}

So, after substituting the value of all the dimensions in the formula we get,

V = \frac{\frac{4}{3} π r_{1}^{3}}{\frac{4}{3} π r_{2}^{3}}

\Rightarrow V = \frac{3^{3}}{5^{3}}

\Rightarrow V = \frac{27}{125}

Therefore, the ratio of the volumes is

27 : 125

.
Hence option (4) is correct.

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