Calculate total ground area required

The ground space of a conical tent corresponds to its circular base area. Since $30$ persons each require $5\text{ m}^2$ ground space, the base area $A$ is:

$A = 30 \times 5 = 150\text{ m}^2$

Calculate total volume required

Each person requires $120\text{ m}^3$ of air to breathe. Therefore, the total volume $V$ of the cone must accommodate all $30$ persons:

$V = 30 \times 120 = 3600\text{ m}^3$

Apply formula for the volume of a cone

The formula relating total volume $V$, base area $A$, and height $h$ of a cone is given by $V = \frac{1}{3} A h$. Substituting $V = 3600\text{ m}^3$ and $A = 150\text{ m}^2$:

$3600 = \frac{1}{3} \times 150 \times h$

Solve for height h

Simplifying the equation to find $h$ gives:

$3600 = 50 \times h \implies h = \frac{3600}{50} = 72\text{ m}$