Question
A cone, a hemisphere, and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio $1 : 2 : 3.$
✓
Answer
Given that, A cone, a hemisphere and a cylinder stand on one equal bases and have the same weight.
We know that $v_{\text {cone }} v _{\text {hemisphere }} v _{\text {Cylinder }} \frac{1}{3} \pi r ^2 h: \frac{2}{3} \pi r ^3: \pi r ^2 h$
Multiplying by $3 \pi r ^2 h: 2 \pi r ^3: 3 \pi r ^2 h$
$\pi r ^3: 2 \pi r ^3: 3 \pi r ^3\left(\therefore r = h\right.$ and $\left.r ^2 h= r ^3\right)$
$1: 2: 3$ Therefore the ratio is $1: 2: 3$.
We know that $v_{\text {cone }} v _{\text {hemisphere }} v _{\text {Cylinder }} \frac{1}{3} \pi r ^2 h: \frac{2}{3} \pi r ^3: \pi r ^2 h$
Multiplying by $3 \pi r ^2 h: 2 \pi r ^3: 3 \pi r ^2 h$
$\pi r ^3: 2 \pi r ^3: 3 \pi r ^3\left(\therefore r = h\right.$ and $\left.r ^2 h= r ^3\right)$
$1: 2: 3$ Therefore the ratio is $1: 2: 3$.
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