A transmitting antenna of height 20 m and the receiving antenna of height h are separated by a distance of 40km for satisfactory communication in line of sight mode. Then the value of h is

Options

A. 40m
B. 45m
C. 30m
D. 25m

Explanation

To solve the problem of finding the height \( h \) of the receiving antenna for satisfactory communication in line of sight mode, we can use the following steps:

### Step 1: Understand the formula for line of sight communication
The formula for the distance \( d \) between two antennas in line of sight mode is given by:
\[ d = \sqrt{2 h_r R} + \sqrt{2 h_t R} \]
where:
- \( d \) is the distance between the antennas,
- \( h_r \) is the height of the receiving antenna,
- \( h_t \) is the height of the transmitting antenna,
- \( R \) is the radius of the Earth (approximately \( 6400 \times 10^3 \) meters).

### Step 2: Substitute known values into the formula
Given:
- \( h_t = 20 \) m (height of the transmitting antenna),
- \( d = 40 \) km = \( 40 \times 10^3 \) m,
- \( R = 6400 \times 10^3 \) m.

Substituting these values into the formula:
\[ 40 \times 10^3 = \sqrt{2 h_r (6400 \times 10^3)} + \sqrt{2 (20)(6400 \times 10^3)} \]

### Step 3: Simplify the equation
Calculate the second term:
\[ \sqrt{2 \times 20 \times 6400 \times 10^3} = \sqrt{256000 \times 10^3} = 16 \times 10^3 \]
Now, the equation becomes:
\[ 40 \times 10^3 = \sqrt{2 h_r (6400 \times 10^3)} + 16 \times 10^3 \]

### Step 4: Isolate the square root term
Rearranging the equation gives:
\[ \sqrt{2 h_r (6400 \times 10^3)} = 40 \times 10^3 - 16 \times 10^3 \]
\[ \sqrt{2 h_r (6400 \times 10^3)} = 24 \times 10^3 \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides results in:
\[ 2 h_r (6400 \times 10^3) = (24 \times 10^3)^2 \]
Calculating the right side:
\[ (24 \times 10^3)^2 = 576 \times 10^6 \]
So, we have:
\[ 2 h_r (6400 \times 10^3) = 576 \times 10^6 \]

### Step 6: Solve for \( h_r \)
Now, isolate \( h_r \):
\[ h_r = \frac{576 \times 10^6}{2 \times 6400 \times 10^3} \]
Calculating the denominator:
\[ 2 \times 6400 \times 10^3 = 12800 \times 10^3 \]
Thus:
\[ h_r = \frac{576 \times 10^6}{12800 \times 10^3} \]
Simplifying:
\[ h_r = \frac{576}{12.8} \text{ m} \]
Calculating this gives:
\[ h_r = 45 \text{ m} \]

### Conclusion
The height of the receiving antenna \( h \) is \( 45 \) meters.

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