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If the radius of a sphere is doubled, what is the ratio of the volumes of the first sphere to that of the second sphere?
Given:
The radius of a sphere is doubled.
To do:
We have to find the ratio of the volumes of the first sphere to that of the second sphere.
Solution:
Let $r$ be the radius of the given sphere.
This implies,
Volume of the sphere $=\frac{4}{3} \pi r^3$
The radius of the new sphere $= 2r$
Therefore,
Volume of the new sphere $=\frac{4}{3} \pi(2 r)^{3}$
$=\frac{4}{3} \pi \times 8 r^{3}$
$=8(\frac{4}{3} \pi r^{3})$
Ratio of the volume of the original sphere and the new sphere $=\frac{4}{3} \pi r^{3}: 8 \times \frac{4}{3} \pi r^{3}$
$=1: 8$
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