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

Options

A. `1:3`
B. `1: sqrt3`
C. `1:2`
D. none of these

Answer

B

Explanation

Let the radius of each of the cone and the hemisphere be r ,Let the height of the cone be h and its slant height be l.
Then ,curved surface area of the hemsisphere
=curved surface area of the cone
`rArr 2 pir^2 = pi rl rArr2pi r^2= pirsqrt(r^2+h^2)`
`rArr = sqrt(r^2+h^2)rArr4r^2+h^2`
[squaring both sides]
`rArr 3r^2 = h^2 rArr (r^2)/(h^2)=1/3rArrr/h=sqrt(1/3)=1/(sqrt(3))`
Hecne , the required ratio is` 1 : sqrt(3)`