# A cone and a cylinder are having the same base . Find the ratio of their heights if their volumes are equal .

Solution:  Let the radius of the common base be $r$ . Let $h_{1}$ and $h_{2}$ be the heights of the cone and cylinder respectively . Then ,

Volume of the cone

$=\frac{1}{3}\pi r^2 h_{1}$ ,

Volume of the cylinder $=\pi r^2h_{2}$

It is given that the cone and the cylinder are of the same volume .

$\therefore$              $\frac{1}{3} \pi r^2 h_{1}=\pi r^2 h_{2}\Rightarrow \frac{1}{3}h_{1}=h_{2}$

$\Rightarrow$           $\frac{h_{1}}{h_{2}}=\frac{3}{1}\Rightarrow h_{1}:h_{2}=3:1$

Hence , the ratio of the height of the cone and cylinder is 3:1.

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