The horizontal distance between two towers is `60` metres. The angle of depression of the top of the first tower when seen from the top of the second tower is `30^(@)` . If the height of the second tower is `90`metres. Find the height of the first tower. [Use `sqrt(3)= 1.732`]

Explanation

To find the height of the first tower, we can follow these steps:

### Step 1: Understand the problem
We have two towers. The height of the second tower (Tower 2) is given as 90 meters. The horizontal distance between the two towers is 60 meters. The angle of depression from the top of Tower 2 to the top of Tower 1 is 30 degrees.

### Step 2: Set up the diagram
Let's denote:
- Height of Tower 1 = \( h_1 \)
- Height of Tower 2 = 90 meters
- Horizontal distance between the two towers = 60 meters
- Angle of depression = 30 degrees

### Step 3: Identify the triangle
When looking from the top of Tower 2 to the top of Tower 1, we can form a right triangle where:
- The vertical side (perpendicular) is the difference in height between the two towers: \( 90 - h_1 \)
- The horizontal side (base) is the distance between the two towers: 60 meters.

### Step 4: Use the tangent function
From the triangle, we can use the tangent of the angle of depression:
\[ \tan(30^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{90 - h_1}{60} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \).

### Step 5: Set up the equation
Substituting the value of \( \tan(30^\circ) \) into the equation gives:
\[ \frac{1}{\sqrt{3}} = \frac{90 - h_1}{60} \]

### Step 6: Cross-multiply to solve for \( h_1 \)
Cross-multiplying gives:
\[ 60 \cdot 1 = \sqrt{3} \cdot (90 - h_1) \]
This simplifies to:
\[ 60 = 90\sqrt{3} - h_1\sqrt{3} \]

### Step 7: Rearranging the equation
Rearranging the equation to isolate \( h_1 \):
\[ h_1\sqrt{3} = 90\sqrt{3} - 60 \]
\[ h_1 = \frac{90\sqrt{3} - 60}{\sqrt{3}} \]

### Step 8: Simplifying the expression
Now, we can simplify \( h_1 \):
\[ h_1 = 90 - \frac{60}{\sqrt{3}} \]
To simplify \( \frac{60}{\sqrt{3}} \), we can multiply the numerator and denominator by \( \sqrt{3} \):
\[ \frac{60}{\sqrt{3}} = \frac{60\sqrt{3}}{3} = 20\sqrt{3} \]

### Step 9: Substitute the value of \( \sqrt{3} \)
Using \( \sqrt{3} \approx 1.732 \):
\[ h_1 = 90 - 20 \times 1.732 \]
Calculating \( 20 \times 1.732 \):
\[ 20 \times 1.732 = 34.64 \]
Thus, we have:
\[ h_1 = 90 - 34.64 = 55.36 \]

### Step 10: Final height of Tower 1
The height of Tower 1 is approximately:
\[ h_1 \approx 55.36 \text{ meters} \]

### Summary
The height of the first tower is approximately 55.36 meters.

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