Discussion
Raju has a cone, a hemisphere and a cylinder. They have the same height. Raju immerses them completely in a bucket full of water. What will be ratio of volume of cylinder to cone to hemisphere, if they also have same bases?
Answer: B
The hemisphere's height is its own radius, so "same height" forces \( h = r \) for all three.
\[ V_{\text{cylinder}} = \pi r^{2} h = \pi r^{3} \]
\[ V_{\text{cone}} = \frac{1}{3}\pi r^{2} h = \frac{1}{3}\pi r^{3} \]
\[ V_{\text{hemisphere}} = \frac{2}{3}\pi r^{3} \]
\[ \text{ratio} = 1 : \frac{1}{3} : \frac{2}{3} = 3 : 1 : 2 \]
This is the classic cylinder–cone–hemisphere relation: the cone is a third of the cylinder and the hemisphere two thirds, so cone plus hemisphere exactly fills the cylinder.
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