The ratio of the curved surfaces areas of a solid right circular cone and of a right circular cylinder both of same radius of base and of same height is 5 : 8 . Then find the ratio of their radius of base and height .

Explanation

Let radius of base = r unit
and height of them = h unit
Also, let the slant height of the cone = l unit
` :.` curved surface of the cone = ` pirl ` wq- unit
` = pir sqrt(r^(2)+h^(2)) `sq - unit
The curved surface area of the cylinder = `2pirh ` sq - unit
As per question , ` (prsqrt(pi^(2)+h^(2)))/(2prh) = 5/8 `
or , ` (sqrt(r^(2) +h^2))/h = 5/4 or, (r^(2)+h^(2))/(h^(2)) = 25/16 or , (r^(2))/(h^(2))+1 = 25/16`
` or , (r^(2))/(h^(2)) = 25/16 - 1 or , (r^(2))/(h^(2)) = 9/16 or , r/h = sqrt(9/16 ) = 3/4`
` rArr r : h = 3 : 4 `
` rArr r: h = 3 :4 `
Hence the required ratio of the radius of base and the height = 3:4