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

In general if we have cone with radius r and height h then area of base of cone is $\pi {{r}^{2}}$(because base of cone is circular) and volume of cone is $\dfrac{1}{3}\pi {{r}^{2}}h$ .

Complete step-by-step answer:
Let radius of the conical tent is r m and the height of the tent is h m.
Space used by each person is 4 sq. meters.
Hence space used by 11 person is $11\times 4=44\,{{m}^{2}}$
So area of base of conical tent is $44\,{{m}^{2}}$
Hence we can write
$\Rightarrow \pi {{r}^{2}}=44$
 $\Rightarrow \dfrac{22}{7}\times {{r}^{2}}=44$
$\Rightarrow {{r}^{2}}=\dfrac{44\times 7}{22}$
$\Rightarrow {{r}^{2}}=14$
$\Rightarrow r=\sqrt{14}m$ ………………………………………(i)
Air required to breathe for one person is 20 ${m}^{3}$.
So area required to breather for 11 person is $11\times 20=220{{m}^{3}}$
So the volume of the conical tent is 220 ${m}^{3}$.
$\Rightarrow \dfrac{1}{3}\pi {{r}^{2}}h=220$
On substituting value of r from equation (i)
$\Rightarrow \dfrac{1}{3}\times \pi \times {{\left( \sqrt{14} \right)}^{2}}\times h=220$
$\Rightarrow \dfrac{1}{3}\times \dfrac{22}{7}\times 14\times h=220$
$\Rightarrow \dfrac{1}{3}\times \dfrac{22}{7}\times 14\times h=220$
$\Rightarrow h=\dfrac{220\times 3\times 7}{22\times 14}$
$\Rightarrow h=15\,m$
Hence the height of the cone is 15m.

Note: In question we have space occupied by one person and air required to breather by one person is given. So don’t forget to multiply by total number of people to get total area and total volume.