Question

$\boxed{ If \:the \:radius \: of\: a \:sphere \:is\: doubled\: then \:what\: is \:the \:ratio\: of \:their\: volumes\:? }$
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Answer 1

Answer :

›»› The ratio of their volumes = 1:8

Given :

• The radius of a sphere is doubled.

To Find :

• The ratio of their volumes = ?

How to Find?

Here in this question we have to find the ratio of their volumes. So, firstly we have to assume the original radius be "r". And here we are provided that The radius of a sphere is doubled then our new radius become 2r So this will give us a new volume also, after this we will find ratio of their volumes on the basis of conditions given above.

Required Solution :

Let ,

The original ratio be "r"

→ Volume of sphere = 4/3 πr³

→ When radius is doubled, then

→ New radius = 2r

→ New volume = 4/3 π(2r)³

→ New volume = 4/3 π (8r³)

Now ,

$\tt {: \implies Ratio = \dfrac{\cancel{\frac{4}{3}}\pi {r}^{3} }{ \cancel{\frac{4}{3}}\pi(8 {r}^{3}) }}$

$\tt {: \implies Ratio = \dfrac{\cancel{\: \pi\:} {r}^{3} }{\cancel{\; \pi\:}(8 {r}^{3}) }}$

$\tt {: \implies Ratio = \dfrac{\cancel{{r}^{3}} }{8\cancel{{r}^{3}} }}$

$\tt {: \implies Ratio = \dfrac{1}{8}}$

$\bf{: \implies Ratio = 1:8}$

║Hence, the ratio of their volumes is 1:8.║