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const2013 [10]
3 years ago
13

Find the surface area of a sphere that has a volume of 288 cu. in.

Mathematics
1 answer:
LenKa [72]3 years ago
4 0

The formula of a volume of a sphere:

V=\dfrac{4}{3}\pi R^3

R - radius

We have the volume = 288 in³. Substitute:

\dfrac{4}{3}\pi R^3=288\qquad\text{multiply both sides by 3}\\\\4\pi R^3=864\qquad\text{divide both sides by}\ 4\pi\\\\R^3=\dfrac{216}{\pi}\to R=\sqrt[3]{\dfrac{216}{\pi}}\\\\R=\dfrac{\sqrt[3]{216}}{\sqrt[3]{\pi}}\\\\R=\dfrac{6}{\sqrt[3]{\pi}}\ in

The formula of a surface area of a sphere:

S.A.=4\pi R^2

Substitute:

S.A.=4\pi\left(\dfrac{6}{\sqrt[3]{\pi}}\right)^2=4\pi\cdot\dfrac{6^2}{\sqrt[3]{\pi^2}}=\dfrac{4\pi\cdot36}{\sqrt[3]{\pi^2}}=\dfrac{144\pi}{\sqrt[3]{\pi^2}}\\\\S.A.=\dfrac{144\pi}{\sqrt[3]{\pi^2}}\cdot\dfrac{\sqrt[3]{\pi}}{\sqrt[3]{\pi}}=\dfrac{144\pi\sqrt[3]{\pi}}{\sqrt[3]{\pi^3}}=\dfrac{144\pi\sqrt[3]{\pi}}{\pi}=\boxed{144\sqrt[3]{\pi}\ in^2}

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