What should be the height of the conical tent ?

Question :

In a marriage ceremony of her daughter Poonam, Ashok has to make arrangements for the accommodation of 150 persons. For this purpose, he plans to build a conical tent in such a way that each person has 4 sq. meters of the space on ground and 20 cubic meters of air to breath. What should be the height of the conical tent ?

Solution :

Let the height of the conical tent = h metre.

Radius of the base of the cone = r meter.

The tent has to accommodate 150 persons.

The space required by each person on the ground = 4 $$m^2$$

And the amount of air = 20 $$m^3$$

$$\therefore$$  Area of the base = 150 $$\times$$ 4 = 600 $$m^2$$

$$\implies$$   $$\pi r^2$$ = 600 $$\implies$$ r = 13.817 m

Volume of the air required for 150 persons = 150 $$\times$$ 20 = 3000 $$m^3$$

$$\implies$$  $$1\over 3$$ $$\pi r^2 h$$ = 3000 $$m^3$$

$$\implies$$ h = $$3000 \times 7 \times 3\over 22 \times (13.817)^2$$ = 15 m

Hence the height of the conical tent is 15 m.