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.

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