Question

in a marriage ceremony of her daughter reena, rajendra has to make accommodation of 150 persons, for this he plans to build a conical tent in such a way that each person has 4m square of space on ground and 20 m cube of air to breathe. What should be height of conical tent?​

Answer 1

Let the height of the conical tent = h m .

Then,the radius of the base of the cone = r m

• The tent has to accommodate = 150 Persons [ Given ]

• The Space required by each person on ground = 4 m² [ Given ]

• Amount of air = 20 m³ [ Given ]

Firstly, We have to find out area of the base :

Area of the base = 15 × 4 = 600 m²

$\implies \sf{\pi {r}^{2} = 600 }$

$\implies \sf{ \dfrac{22}{7} \times {r}^{2} = 600 }$

$\implies \sf{ {r}^{2} = \dfrac{600 \times 7}{22} }$

$\implies \sf{ {r}^{2} = \dfrac{4200}{22} }$

$\implies \sf{ {r}^{2} = 190.9.}$

$\implies \sf{r = 13.817 \: m}$

Now,

Volume of the air required for 150 Persons = 150 × 20 = 3000 m³

$\implies \sf{ \dfrac{1}{3}\pi {r}^{2}h = 3000 }$

$\implies \sf{ \dfrac{1}{3} \times \dfrac{22}{7} \times {(13.817)}^{2} \times h = 3000 }$

$\implies \sf{h = \dfrac{3000 \times 7 \times 3}{22 \times {(13.817)}^{2} } }$

$\implies \sf{ h = \cancel \dfrac{63,000}{4,200} }$

$\implies \sf{ h = 15 \: m}$

Hence, the height of the conical tent is 15 m.

Answer 2

Let the area of the base of conical tent be ,

⇒ Total people * Gap b/w the persons

⇒ 150 * 4

⇒ 600 m²

Now base of the cone resembles the circle.

⇒ πr² = 600 m² ... (1)

Volume of the air required for 1 person = 20 m³

So volume of air required for 150 persons = 150 * 20 = 3000 m³

We can find the height of the conical tent through the volume of cone.

$\underbrace{\bold{Volume\ of\ cone\ , V=\frac{1}{3}\pi r^2h }}$

⇒ 1/3 * πr²h = 3000

⇒ 1/3 * ( πr² ) h = 3000

⇒ 1/3 * ( 600 ) h = 3000 [ From (1) ]

⇒ h/3 = 5

⇒ h = 15 meters

So , height of the conical tent should be 15 meters.