Question

if the radius of a sphere is halved find the ratio of the volume of the original solid to the volume of the resulting solid

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Answer 1

Answer:

8 times

Step-by-step explanation:

volume of sphere=4/3πr^3

when radius is halved,

volume of sphere=4/3π(r/2)^3

ratio=(4/3πr^3)/(4/3π(r/2)^3)

=r^3/(r/2)^3

=r^3/r^3/8

=8/1

ratio=8:1

Answer 2

Ratio of the volume of the original solid to the volume of the resulting solid is 8 : 1

Solution:

Volume of sphere is given as:

$Volume\ of\ sphere = \frac{4}{3} \pi r^3 -------- eqn\ 1$

Where, "r" is the radius of sphere

The radius of a sphere is halved

$r = \frac{r}{2}$

Volume of resulting sphere is given as:

$Volume\ of\ resulting\ sphere = \frac{4}{3} \pi (\frac{r}{2})^3\\\\Volume\ of\ resulting\ sphere = \frac{4}{3} \times \pi \times \frac{r^3}{8}\\\\Volume\ of\ resulting\ sphere = \frac{\pi r^3}6} ------- eqn\ 2$

Find the ratio of the volume of the original solid to the volume of the resulting solid

Divide eqn 1 by eqn 2

$Ratio = \frac{4}{3} \times \pi r^3 \div \frac{\pi r^3}{6}\\\\Ratio = \frac{4}{3} \times \pi r^3 \times \frac{6}{\pi r^3}\\\\Ratio = \frac{8}{1}$

Thus ratio of the volume of the original solid to the volume of the resulting solid is 8 : 1

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