Question

The surface areas of two spheres are in the ratio 1:4. Find the ratio of their volumes.

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Answer 1

Heya !

Given:-

• Ratios of the S.A of two spheres = 1:4

To find :-

• ratio of the Volumes of both the Spheres.

Fornula to be used. :-

• $\huge \fbox \blue{S.A of sphere = 4πr²}$

• $\huge \fbox \blue{Volume of sphere = 4/3πr³}$

Solution:-

• let us assume radius of sphere 1 as r and radius of sphere 2 as R.

acc. to d question:-

1/4 = S.A of sphere 1/ S.A of sphere 2

1/4 = 4πr²/4πR²

1/4 = r²/R²

(1/2)² = (r/R)²

hence, r/R = 1/2 ...........eq. 1

now , ratio of their volumes ;

$\frac{ \frac{4}{3} \times \pi {r}^{3} }{ \frac{4}{3} \times \pi {R}^{3} } = \frac{ {r}^{3} }{ {R}^{3} } = { \frac{r}{R} }^{3}$

Value of r/R is 1/2 ( from eq. 1)

therefore, (r/R)³ = (1/2)³ = 1/8.

Hence, the ratio becomes 1:8.