A transmitting antenna at the top of a tower has a height 32 m and the height of the receiving antenna is 50 m. What is the maximum distance between them for satisfactory communication in line of sight (LOS) mode?

To find the maximum distance for satisfactory communication in line of sight (LOS) mode between a transmitting antenna of height 32 m and a receiving antenna of height 50 m, we use the formula that accounts for the curvature of the Earth. The maximum LOS distance $$ d $$ is the sum of the individual horizon distances from each antenna:

$$ d = \sqrt{2R h_t} + \sqrt{2R h_r} $$

where:

• $$ R $$ is the radius of the Earth, approximately 6400 km,
• $$ h_t $$ is the height of the transmitting antenna in km,
• $$ h_r $$ is the height of the receiving antenna in km.

First, convert the heights from meters to kilometers:

$$ h_t = 32 \text{m} = \frac{32}{1000} \text{km} = 0.032 \text{km} $$

$$ h_r = 50 \text{m} = \frac{50}{1000} \text{km} = 0.050 \text{km} $$

Substitute the values into the formula:

$$ d = \sqrt{2 \times 6400 \times 0.032} + \sqrt{2 \times 6400 \times 0.050} $$

Calculate each term separately. Start with the first term:

$$ 2 \times 6400 = 12800 $$

$$ 12800 \times 0.032 = 409.6 $$

$$ \sqrt{409.6} = \sqrt{\frac{4096}{10}} = \frac{\sqrt{4096}}{\sqrt{10}} = \frac{64}{\sqrt{10}} \quad (\text{since } 64^2 = 4096) $$

Now the second term:

$$ 12800 \times 0.050 = 640 $$

$$ \sqrt{640} = \sqrt{64 \times 10} = \sqrt{64} \times \sqrt{10} = 8\sqrt{10} $$

Combine both terms:

$$ d = \frac{64}{\sqrt{10}} + 8\sqrt{10} $$

To add these, express them with a common denominator:

$$ 8\sqrt{10} = \frac{8\sqrt{10} \times \sqrt{10}}{\sqrt{10}} = \frac{8 \times 10}{\sqrt{10}} = \frac{80}{\sqrt{10}} $$

$$ d = \frac{64}{\sqrt{10}} + \frac{80}{\sqrt{10}} = \frac{144}{\sqrt{10}} $$

Rationalize the denominator:

$$ d = \frac{144}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{144\sqrt{10}}{10} = \frac{144}{10} \times \sqrt{10} = 14.4\sqrt{10} $$

Now, approximate $$ \sqrt{10} \approx 3.162 $$:

$$ d \approx 14.4 \times 3.162 $$

Calculate step by step:

$$ 14.4 \times 3 = 43.2 $$

$$ 14.4 \times 0.162 = 14.4 \times 0.16 = 2.304 \quad \text{and} \quad 14.4 \times 0.002 = 0.0288 $$

$$ 2.304 + 0.0288 = 2.3328 $$

$$ d \approx 43.2 + 2.3328 = 45.5328 \text{km} $$

Rounding to one decimal place, $$ d \approx 45.5 \text{km} $$.

Comparing with the options, 45.5 km corresponds to option A.

Hence, the correct answer is Option A.