Answer
$12\pi\; in.^2$.
Work Step by Step
The given values are
Volume $V=84\pi\; in.^3$
Height $h=7\;in.$
Area of the base.
$\Rightarrow B=\pi r^2 $
Volume of the cylinder is
$\Rightarrow V=\pi r^2 h$
Substitute all the values.
$\Rightarrow 84\pi\; in.^3=\pi r^2 (7\;in.)$
Divide both sides by $(7\;in.)$.
$\Rightarrow \frac{84\pi\; in.^3}{(7\;in.)}=\frac{\pi r^2 (7\;in.)}{(7\;in.)}$
Cancel common terms.
$\Rightarrow 12\pi\; in.^2=\pi r^2$
$\Rightarrow 12\pi\; in.^2=B$
Hence, the area of the cylinder is $12\pi\; in.^2$.
