Question

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

Answer 1

The formula of a volume of a sphere:

`V=(4)/(3)\pi R^3`

R - radius

We have the volume = 288 in³. Substitute:

`(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=(216)/(\pi)\to R=\sqrt[3]{(216)/(\pi)}\\\\R=\frac{\sqrt[3]{216}}{\sqrt[3]{\pi}}\\\\R=\frac{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(\frac{6}{\sqrt[3]{\pi}}\right)^2=4\pi\cdot\frac{6^2}{\sqrt[3]{\pi^2}}=\frac{4\pi\cdot36}{\sqrt[3]{\pi^2}}=\frac{144\pi}{\sqrt[3]{\pi^2}}\\\\S.A.=\frac{144\pi}{\sqrt[3]{\pi^2}}\cdot\frac{\sqrt[3]{\pi}}{\sqrt[3]{\pi}}=\frac{144\pi\sqrt[3]{\pi}}{\sqrt[3]{\pi^3}}=\frac{144\pi\sqrt[3]{\pi}}{\pi}=\boxed{144\sqrt[3]{\pi}\ in^2}`