Question

a cone,a hemisphere and a cylinder stand on equal bases and volumes and the same height .
show that their volumes are in the ratio of 1:2:3

Answer 1

Given :

A cone , hemisphere and a cylinder stand on equal bases

Hence ,,

Radius of base of cone = Radius of base of cylinder = Radius of base of hemisphere = r

And they also have same height = h

Height of a cone = Height of cylinder = Height of hemisphere ( r ) = h ( r is the radius of the sphere )

Volume of cone = 1 / 3 πr^2h --- ( 1 )

Volume of hemisphere = 2 / 3 πr^3 = 2 / 3 πr^2h ( r = h in hemisphere) --- ( 2 )

Volume of Cylinder = πr^2h --- ( 3 )

Eq ( 1 ) : Eq ( 2 ) : Eq ( 3 )

$= > V_{c} : V_{h} : V_{cy} \\ \\ = > \frac{1}{3} \pi {r}^{2} h : \frac{2}{3} \pi {r}^{2} h : \pi {r}^{2} h \\ \\ = > \frac{1}{3} : \frac{2}{3} : 1 \\ => 1:2:3$

Hence proved ....

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