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.

Given,

Each person must have 16 m³ of air to breathe.

∴ 77 persons need 77 × 16 m³ = 1232 m³

Radius of the tent (r) = 7 m

Let height of the conical tent be h meters.

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

$\Rightarrow \dfrac{1}{3}π \text{r}^2 \text{h} = 1232 \\[1em] \Rightarrow \dfrac{1}{3} \times \dfrac{22}{7} \times 7^2 \times \text{h} = 1232 \\[1em] \Rightarrow \dfrac{1}{3} \times \dfrac{22}{7} \times 49 \times \text{h} = 1232 \\[1em] \Rightarrow \text{h} = \dfrac{1232 \times 7 \times 3}{22 \times 49} \\[1em] \Rightarrow \text{h} = \dfrac{25872}{1078} \\[1em] \Rightarrow \text{h} = 24 \text{ m.}$

By formula,

l² = r² + h²

⇒ l² = 7² + 24²

⇒ l² = 49 + 576

⇒ l² = 625

⇒ l = $\sqrt{625}$ = 25 m

Curved surface area of the tent = πrl

$= \dfrac{22}{7} \times 7 \times 25 \\[1em] = 22 \times 25 \\[1em] = 550 \text{ m}^2$

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