A conical tent is to accommodate 11 persons . Each persons must have 4 sq. metres of the space on the ground and 20 cubic metres of air to breath . Find the height of the cone .

Solution:  Let $h$ metres be the height , $r$ metres the radius of base of the cone . Since the tent can accommodate 11 persons and each person requires 4 sq. metres of the space on the ground and 20 cubic metres of air . Therefore,

Area of the base $=(11\times4)m^2=44m^2\Rightarrow \pi r^2=44m^2......................(1)$

Volume of the cone $=(11\times20)m^3=220m^3\Rightarrow \frac{1}{3}\pi r^2h=220m^3.................(2)$

Dividing (2) by (1) , we get

$\frac{\frac{1}{3}\pi r^2h}{\pi r^2}=\frac{220}{44}\Rightarrow \frac{h}{3}=5\Rightarrow h=15m.$

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