What should be the height of transmitting antenna and the...

What should be the height of transmitting antenna and the population covered if the television telecast is to cover a radius of

150km ?

The average population density around the tower is

2000{km}^{- 2}

and the value of

{R}_{e} = 6 . 5 \times {10}^{6}m .

Options

AHeight $= 1731m$
Population Covered $= 1413 \times {10}^{5}$
BHeight $= 1241m$
Population Covered $= 7 \times {10}^{5}$
CHeight $= 1600m$
Population Covered $= 2 \times {10}^{5}$
DHeight $= 1800m$
Population Covered $= 1413 \times {10}^{8}$

Correct answer

A. Height $= 1731m$
Population Covered $= 1413 \times {10}^{5}$

Step-by-step solution

Radius covered, $r = \sqrt{2R{H}_{T}}$
$150km = \sqrt{2 \times \left(6 . 5 \times {10}^{6}m\right){H}_{T}}$
${\left(150km \times {10}^{3}\right)}^{2} = 2 \times 6 . 5 \times {10}^{6}{H}_{T}$
${H}_{T} = 1731m$

Population covered $= \left(\pi{r}^{2}\right)\left(2000{km}^{- 2}\right)$
$= 3 . 14 \times ( 150{)}^{2} \times 2000 = 1413 \times {10}^{5}$

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