Question

The sum of the inner and the outer curved surface of a hollow cylinder is $1056$ $c{m}^2$ and the volume of material in it is $1056$ $c{m}^3$. Find sum of its internal radii. Given that the height of the cylinder is $21$ cm.

Answer 1

Let us take external radii be $R$
Inner radii $=r$ and height $=21$ cm
As sum of the inner and the outer curved surfaces of a hollow metallic cylinder is $1056c{m}^2$,
$\therefore \pi Rh+\pi rh=1056$
$\therefore R+r=8$ .....(1)
and volume of material $=1056c{m}^3$
$\Rightarrow \pi R^{2}h-\pi r^{2}h=1056$
$\Rightarrow R^2-r^2=16$
$\Rightarrow (R+r)(R-r)=16$
$\Rightarrow R-r=2$ .....(2)
After adding (1) and (2), we get
$2R=10$
$\Rightarrow R=5$
$\Rightarrow r=R-2=5-2=3$
$\therefore$ external radius $R$ is $5$ cm and internal radius $r$ is $3$ cm.