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Q.

A cone, a hemisphere and a cylinder have the same base and same height. Calculate the ratio of their volumes.

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a

1 : 2 : 3

b

1 : 2 : 4

c

1 : 2 : 5

d

1 : 2 : 6

answer is A.

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Detailed Solution

As, cone, hemisphere and cylinder have same base. So,
Radius of cone = Radius of hemisphere = Radius of cylinder = r(say)
Also, cone, hemisphere and cylinder have same height. So,
Height of cone = Height of hemisphere = Height of cylinder = h(say)
Now,
Volume of cone =

\frac{1}{3} π r^{2} h

Volume of hemisphere =

\frac{2}{3} π r^{3}

Volume of cylinder

π r^{2} h

So, the required ratio,
cone: hemisphere: cylinder

= \frac{1}{3} π r^{2} h : \frac{2}{3} π r^{3} : π r^{2} h

= \frac{1}{3} π r^{2} r : \frac{2}{3} π r^{3} : π r^{2} r

= \frac{1}{3} π r^{3} : \frac{2}{3} π r^{3} : π r^{3}

= \frac{1}{3} : \frac{2}{3} : 1

= 1 : 2 : 3

Correct option is 1.

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