Question

A solid cylinder has total surface area of $48\pi c{m}^2$ Its curved surface area is $\frac{5}{8}$ times of its total surface area. Find the radius and the height of the cylinder.

Answer 1

Let r and h be the radius and height of the cylinder.

$\therefore$ Total surface area $=48\pi c{m}^2$

$2\pi rh+2\pi r^2=48\pi$

$2\pi (rh+r^2)=48\pi$

$\Rightarrow rh+r^2=\frac{48\pi }{2\pi }$

$\Rightarrow rh+r^2=24$ …………(i)

and curved surface area $=\frac{5}{8}$ total surface area

$\Rightarrow 2\pi rh=\frac{5}{8}.48\pi$

$\Rightarrow 2\pi rh=30\pi$

$\Rightarrow rh=\frac{30\pi }{2\pi }$

$\Rightarrow rh=15$ …………….(ii)

From (i),

$15+r^2=24$

$\Rightarrow r^2=24-15$

$\Rightarrow r^2=9$

$\Rightarrow r=3$cm

From (ii)

$h=\frac{15}{r}$

$h=\frac{15}{3}$

$h=5$cm

Hence radius $=3$cm

Height $=5$cm.