A TV transmission tower has a height of 140 m and the height of the receiving antenna is 40 m. What is the maximum distance upto which signals can be broadcasted from this tower in LOS (Line of Sight) mode? (Given: radius of earth $$= 6.4 \times 10^6$$ m).

For propagation in the LOS (Line of Sight) mode, the transmitting antenna at height $$h_1$$ can “see” up to the horizon that lies at a distance $$d_1$$, and the receiving antenna at height $$h_2$$ can “see” back toward the transmitter up to a distance $$d_2$$.

The standard geometric result for each horizon distance (measured along the Earth’s surface) is first stated:

$$d \;=\;\sqrt{2Rh}$$

where $$R$$ is the radius of the Earth and $$h$$ is the height of the antenna in metres.

We have

$$R \;=\;6.4 \times 10^{6}\text{ m}, \qquad h_1 \;=\;140\text{ m}, \qquad h_2 \;=\;40\text{ m}. $$

Applying the formula to the transmitting tower:

$$d_1 = \sqrt{2Rh_1} = \sqrt{2 \times (6.4 \times 10^{6}) \times 140}. $$

First multiply the two numbers inside the root:

$$2 \times 6.4 \times 10^{6} = 1.28 \times 10^{7},$$

so

$$1.28 \times 10^{7} \times 140 = 1.792 \times 10^{9}.$$

Hence

$$d_1 = \sqrt{1.792 \times 10^{9}} = \sqrt{1.792}\;\sqrt{10^{9}} = 1.338 \times 3.162 \times 10^{4}\text{ m} \approx 4.230 \times 10^{4}\text{ m} = 4.23 \times 10^{4}\text{ m}. $$

Converting this to kilometres:

$$d_1 \approx 42.3\text{ km}. $$

Now, applying the same formula to the receiving antenna:

$$d_2 = \sqrt{2Rh_2} = \sqrt{2 \times (6.4 \times 10^{6}) \times 40}. $$

Compute the product inside the root:

$$1.28 \times 10^{7} \times 40 = 5.12 \times 10^{8}.$$

Thus

$$d_2 = \sqrt{5.12 \times 10^{8}} = \sqrt{5.12}\;\sqrt{10^{8}} = 2.262 \times 10^{4}\text{ m} = 2.262 \times 10^{4}\text{ m}. $$

In kilometres this is

$$d_2 \approx 22.6\text{ km}. $$

The total maximum LOS distance is the sum of the two horizon distances:

$$d_{\text{max}} = d_1 + d_2 \approx 42.3\text{ km} + 22.6\text{ km} = 64.9\text{ km}. $$

Rounding appropriately,

$$d_{\text{max}} \approx 65\text{ km}. $$

Hence, the correct answer is Option D.