The horizontal distance between two towers is 75 m and the angular depression of the top of the first tower as see from the top of the second, which is 160 m high, is `45^(@)` . Find the height of the first tower.

Options

A. `80` m
B. `81` m
C. `85` m
D. None

Answer

To solve the problem step by step, we will use the information provided in the question and apply the concepts of trigonometry.

### Step 1: Understand the Problem
We have two towers, Tower A (the first tower) and Tower B (the second tower). The height of Tower B (160 m) and the horizontal distance between the two towers (75 m) are given. The angle of depression from the top of Tower B to the top of Tower A is 45 degrees. We need to find the height of Tower A.

### Step 2: Draw a Diagram
Draw a diagram to visualize the problem. Label the points:
- Point A: Top of Tower A
- Point B: Top of Tower B
- Point C: Base of Tower A
- Point D: Base of Tower B

The horizontal distance between points C and D (the bases of the two towers) is 75 m. The height of Tower B (point B) is 160 m. The angle of depression from point B to point A is 45 degrees.

### Step 3: Identify the Triangle
From point B (top of Tower B), draw a line to point A (top of Tower A). This line forms a right triangle with the horizontal line from point B to point D (base of Tower B) and the vertical line from point D to point A (height of Tower A).

### Step 4: Apply Trigonometry
In triangle BDA:
- The angle of depression is 45 degrees.
- The height of Tower B (BD) is 160 m.
- The horizontal distance (CD) is 75 m.

Since the angle of depression is equal to the angle of elevation from point A to point B, we can use the tangent function:
\[ \tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{75} \]
Where \( h \) is the height of Tower A (CD).

### Step 5: Calculate the Height
Since \(\tan(45^\circ) = 1\):
\[ 1 = \frac{h}{75} \]
Cross-multiplying gives:
\[ h = 75 \]

### Step 6: Find the Height of Tower A
Now, we need to find the height of Tower A (CD). The total height of Tower B (160 m) minus the height from point D to point A (75 m) gives us:
\[ CD = 160 - 75 = 85 \text{ m} \]

### Conclusion
The height of the first tower (Tower A) is 85 meters.