Question

A right circular cylinder and a right circular cone have equal bases and equal heights. If their curved surfaces are in the ratio 8 : 5, determine the ratio of the radius of the base to the height of either of them.

Answer 1

For right circular cylinder, let r₁ = r, h₁ = h.

Then, curved surface area, s1 of cylinder = `2pir_1 = 2pirh   ........... (1)`

For right circular cone, let r₂ = r, h₂ = h

Then, curved surface area, s2 of cone = `pi r_2l " where "l = sqrt(r_2^2 + h_2^2) = sqrt(r^2 + h^2)`

                                                           ` = pir sqrt(r^2 + h^2)    ..................(2)`

Divide (i) and (ii),

`s_1/s_2 = (2pirh)/(pirsqrt(r^2 + h^2))`

`8/5 = (2h)/sqrt(r^2 + h^2)    [s_1/s_2 = 8/5]`

`64/25 = (4h^2)/(r^2 + h^2)` [squaring]

`64r^2 + 64h^2 = 100h^2`

`64r^2 = 36h^2`

`16r^2 = 9h^2`

`r^2 / h^2 = 9/16`

`r/h = 3/4`

`therefore r : h = 3:4`