A conical tent is to accommodate 11 persons. Each person must have 4 m² of the space on the ground and 20m³ of air to breathe. Find the height of the cone.

Given,

Each person must have 20 m³ of air to breathe.

∴ 11 persons need 11 × 20 m³ = 220 m³

Each person must have 4 m² of the space on the ground.

∴ 11 persons need 11 × 4 m² = 44 m²

Base of the conical tent = area of the circle = πr²

$\Rightarrow 44 = \dfrac{22}{7} \times \text{r}^2 \\[1em] \Rightarrow \text{r}^2 = \dfrac{7 \times 44}{22} \\[1em] \Rightarrow \text{r}^2 = \dfrac{308}{22} \\[1em] \Rightarrow \text{r}^2 = 14 \text{ m.}$

Let height of the conical tent be h meters.

Since, conical tent needs to accomodate 11 persons, so its volume will be equal to volume of air required for 11 persons.

$\Rightarrow \dfrac{1}{3}π \text{r}^2 \text{h} = 220 \\[1em] \Rightarrow \dfrac{1}{3} \times \dfrac{22}{7} \times 14 \times \text{h} = 220 \\[1em] \Rightarrow \text{h} = \dfrac{220 \times 7 \times 3}{22 \times 14} \\[1em] \Rightarrow \text{h} = \dfrac{4620}{308} \\[1em] \Rightarrow \text{h} = 15 \text{ m.}$

Hence, height of the tent is 15 m..