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A conical tent is to accommodate 77 persons. Each person must have 16 m3 of air to breathe. Given the radius of the tent as 7 m, find the height of the tent and also its curved surface area.

Mensuration

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Answer

Given,

Each person must have 16 m3 of air to breathe.

∴ 77 persons need 77 × 16 m3 = 1232 m3.

Radius of tent (r) = 7 m.

Let height of conical tent be h meters.

Since, conical tent needs to accommodate 77 persons, so its volume will be equal to volume of air required for 77 persons.

⇒13πr2h=1232⇒13×227×72×h=1232⇒13×22×7×h=1232⇒h=1232×322×7⇒h=3696154⇒h=24 m.\Rightarrow \dfrac{1}{3}πr^2h = 1232 \\[1em] \Rightarrow \dfrac{1}{3} \times \dfrac{22}{7} \times 7^2 \times h = 1232 \\[1em] \Rightarrow \dfrac{1}{3} \times 22 \times 7 \times h = 1232 \\[1em] \Rightarrow h = \dfrac{1232 \times 3}{22 \times 7} \\[1em] \Rightarrow h = \dfrac{3696}{154} \\[1em] \Rightarrow h = 24 \text{ m}.

By formula,

⇒ l2 = r2 + h2

⇒ l2 = 72 + 242

⇒ l2 = 49 + 576

⇒ l2 = 625

⇒ l = 625\sqrt{625} = 25 m.

Curved surface area of tent = πrl

= 227×7×25\dfrac{22}{7} \times 7 \times 25

= 22 × 25

= 550 m2.

Hence, height of tent = 24 m and curved surface area of tent = 550 m2.

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