Question

A hemisphere and a cone have equal bases. If their heights are also equal, then what is the ratio of their curved surfaces?

Answer 1

The base of the cone and hemisphere are equal. So radius of the two is also equal.

and

Height of the hemisphere = height of the cone

Then the slant height of the cone

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

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

` = sqrt(2r^2)`

`= rsqrt2     ................ (1)`

Now, the curved surface area of

Hemisphere ` =2pir^2`

and

The curved surface area of cone `=pirl`

Putting the value of l from eq. (i)

We get

`=pirsqrt2  r`

`=pir^2 sqrt2  r `

Now,

`"C .S .A. of hemisphare"/"C.S.A. of cone" = (2pir^2)/(pirsqrt2)`

`=2/sqrt2 xx sqrt2/sqrt2`

`=(2sqrt2)/2`

`=sqrt2 :1`