Question

A solid is in the shape of a cone mounted on a hemisphere of same base radius. If the curved surface areas of the hemispherical part and the conical part are equal, then find the ratio of the radius and the height of the conical part.

Answer 1

Let ABC be a cone, which is mounted on a hemisphere.

Given: OC = OD = r cm

Curved surface area of the hemispherical part

= `1/2 (4πr^2)`

= 2πr²

Slant height of a cone,

`l = sqrt(r^2 + h^2)`

So, curved surface area of a cone = πrl

= `πr sqrt(h^2 + r^2)`

i.e., `2πr^2 = πr sqrt(h^2 + r^2)`   ...(Given)

⇒ `2r = sqrt(h^2 + r^2)`

On squaring both of the sides, we get

4r² = h² + r²

⇒ 4r² – r² = h²

⇒ 3r² = h²

⇒ `r^2/h^2 = 1/3`

⇒ `r/h = 1/sqrt(3)`

Hence, the ratio of the radius and the height = `1 : sqrt(3)`.