The maximum distance between the transmitting and receiving TV towers is 72 km. If the ratio of the heights of the TV transmitting tower to receiving tower is `16:25`, the heights of the transmitting and receiving towers are

Options

A. `51.2` m , 80 m
B. 40 m , 80 m
C. 80 m , 125 m
D. 25 m , 75 m

Answer

To solve the problem of finding the heights of the transmitting and receiving TV towers given the maximum distance and the ratio of their heights, we can follow these steps:

### Step 1: Understand the Given Information
We know:
- The maximum distance \( d \) between the transmitting and receiving towers is 72 km.
- The ratio of the heights of the transmitting tower \( H_T \) to the receiving tower \( H_R \) is \( \frac{H_T}{H_R} = \frac{16}{25} \).

### Step 2: Use the Formula for Maximum Line of Sight
The formula for the maximum line of sight distance between two antennas is given by:
\[ d = \sqrt{2 H_T \cdot R} + \sqrt{2 H_R \cdot R} \]
where \( R \) is the radius of the Earth. For our calculations, we will assume \( R \) is approximately 6400 km (or 6400000 meters).

### Step 3: Express \( H_T \) in Terms of \( H_R \)
From the ratio of the heights:
\[ H_T = \frac{16}{25} H_R \]

### Step 4: Substitute \( H_T \) into the Distance Formula
Substituting \( H_T \) into the distance formula:
\[ 72 = \sqrt{2 \left(\frac{16}{25} H_R\right) \cdot 6400000} + \sqrt{2 H_R \cdot 6400000} \]

### Step 5: Simplify the Equation
Let’s simplify the equation:
1. Calculate \( \sqrt{2 \cdot 6400000} \):
\[ \sqrt{2 \cdot 6400000} = \sqrt{12800000} \approx 3577.4 \]
2. Substitute back into the equation:
\[ 72 = \sqrt{2 \cdot \frac{16}{25} H_R \cdot 6400000} + \sqrt{2 H_R \cdot 6400000} \]
This becomes:
\[ 72 = \frac{16}{25} \cdot 3577.4 \sqrt{H_R} + 3577.4 \sqrt{H_R} \]
Factor out \( \sqrt{H_R} \):
\[ 72 = \left(\frac{16}{25} + 1\right) \cdot 3577.4 \sqrt{H_R} \]
\[ 72 = \left(\frac{16 + 25}{25}\right) \cdot 3577.4 \sqrt{H_R} \]
\[ 72 = \frac{41}{25} \cdot 3577.4 \sqrt{H_R} \]

### Step 6: Solve for \( H_R \)
Now, isolate \( \sqrt{H_R} \):
\[ \sqrt{H_R} = \frac{72 \cdot 25}{41 \cdot 3577.4} \]
Calculating the right side:
\[ \sqrt{H_R} \approx \frac{1800}{146.5} \approx 12.28 \]
Now square both sides to find \( H_R \):
\[ H_R \approx (12.28)^2 \approx 150.7 \text{ meters} \]

### Step 7: Find \( H_T \)
Using the ratio to find \( H_T \):
\[ H_T = \frac{16}{25} H_R = \frac{16}{25} \cdot 150.7 \approx 96.4 \text{ meters} \]

### Conclusion
Thus, the heights of the transmitting and receiving towers are approximately:
- \( H_T \approx 96.4 \) meters
- \( H_R \approx 150.7 \) meters