Question

A TV tower has a height of 150 m. The area of the region covered by TV broadcast is (radius of Earth = $6.4 × 10^6 m$)

A $9.6π × 10^8 m^2$

B $19.2π × 10^8 m^2$

C $19.2π × 10^7 m^2$

D $1.92π × 10^5 km^2$

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Answer

The correct option is $\mathbf{B}$

$$ 19.2 \pi \times 10^{8} , \text{m}^{2} $$

To find the area of the region covered by the TV broadcast, we use the following formula:

$$ \text{Area} = \pi d^{2} $$

Here, $d$ is the diameter of the broadcast region, which is related to the height of the TV tower ($h$) and the radius of the Earth ($R$) by the formula:

$$ d = \sqrt{2hR} $$

Given:

  • Height of the TV tower, $h = 150 , \text{m}$

  • Radius of the Earth, $R = 6.4 \times 10^{6} , \text{m}$

Therefore, the area becomes:

$$ \text{Area} = \pi (2hR) $$

We then substitute the given values:

$$ \begin{aligned} \text{Area} &= \pi (2 \times 150 \times 6.4 \times 10^{6}) , \text{m}^{2} \ &= \pi \times 300 \times 6.4 \times 10^{6} , \text{m}^{2} \ &= 19.2 \pi \times 10^{8} , \text{m}^{2} \end{aligned} $$

Hence, the area covered by the TV broadcast is $ 19.2 \pi \times 10^{8} , \text{m}^{2} $, which corresponds to option B.


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