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 of each is to the height of each as 3:4.

Explanation

Let their curved surfaces be 8x and 5x. Then,
`2pi rh = 8x and pi r sqrt(h^(2) + r^(2)) = 5x`
`rArr 4pi^(2) r^(2) h^(2) = 64x^(2) and pi^(2) r^(2) (h^(2) + r^(2)) = 25x^(2)`
`rArr (4pi^(2) r^(2)h^(2))/(pi^(2) r^(2) (h^(2) + r^(2))) = (64)/(25) rArr 9h^(2) = 16r^(2) rArr (r)/(h) = (3)/(4)`