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