Question

A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio 8:5, show that the radius and height of each has the ratio 3:4.

Answer 1

Let the radius and height of cylinder and cone be r and h.
Let the slant height of cone be l.

CSA of cylinder : $2\pi rh$
CSA of cone : $\pi rl$

Ratio = CSA of cylinder/CSA of cone =$2\pi rh$ $÷$ $\pi rl$= 8/5

⇒ h/l = 4/5
⇒ $h^2/l^2$ = 16/25
⇒l² = (25/16)h²
⇒$h^2+r^2=\left(25/16\right)h^2$
⇒ $r^2=\left(9/16\right)h^2$
⇒ $(r/h{)}^2=(3/4{)}^2$
⇒ r/h = 3/4

hence ratio of radius to height is 3:4