A TV transmission tower has a height of $h=100\text{ m}$. If the average population density around the tower is $\rho =1000{\text{ per km}}^2$, how many people can be covered by the broadcast? (Take radius of Earth $R_e=6400\text{ km}$)

Solution

1. The broadcast range $d$ is given by $d=\sqrt{2R_{e}h}$, where $R_e=6400\text{ km}$ and $h=0.1\text{ km}$.
2. Substituting the values, $$\begin{array}{rl}d& =\sqrt{2\times 6400\times 0.1}\\ & =\sqrt{1280}\approx 35.77\text{ km}\end{array}$$.
3. The coverage area is $$\begin{array}{rl}A& =\pi d^2\\ & =\pi \times 1280\approx 4021{\text{ km}}^2\end{array}$$.
4. The total population covered is $A\times \rho =4021{\text{ km}}^2\times 1000{\text{ people/km}}^2\approx 4.02\text{ million}$.
5. Hence the answer is (C).