Question #9757e

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Question #9757e

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Answer 1 · 2016-05-08T04:01:12

I assume, the area $S = 233 π$ $S=233pi$ is given, not circumference.
Then
$V \approx 4742.12 π$ $V~~4742.12pi$

In terms of given circumference $C$ $C$ ,
$V = \frac{C}{6 π^{2}}$ $V=C/(6pi^2)$

Explanation:

The formula for volume of a sphere of a radius $R$ $R$ is
$V = \frac{4}{3} π R^{3}$ $V=4/3piR^3$

The formula for area of a sphere of the same radius is
$S = 4 π R^{2}$ $S=4piR^2$

Given the surface area, we can find a radius from the last formula:
$R = \sqrt{\frac{S}{4 π}} = \sqrt{\frac{233 π}{π}} = \sqrt{233}$ $R=sqrt(S/(4pi))= sqrt((233pi)/pi)=sqrt(233)$

Now we can fund volume:
$V = \frac{4}{3} π R^{3} = \frac{4}{3} π {(\sqrt{233})}^{3} \approx 4742.12 π$ $V=4/3piR^3=4/3pi(sqrt(233))^3~~4742.12pi$

If, instead of an area, circumference of the equator (the largest circle on a surface of a sphere) is given, the calculations are:
$C = 2 π R$ $C=2piR$
$R = \frac{C}{2 π}$ $R=C/(2pi)$
$V = \frac{4}{3} π R^{3} = \frac{4}{3} π {(\frac{C}{2 π})}^{3} = \frac{4}{3} π \frac{C^{3}}{8 π^{3}} = \frac{C^{3}}{6 π^{2}}$ $V=4/3piR^3=4/3pi(C/(2pi))^3=4/3piC^3/(8pi^3)=C^3/(6pi^2)$

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Question #9757e

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