The horizontal distance between two towers is 60 m and the angle of depression of the top of the first tower as seen from the top to the second is 30. If the height of the second tower is 150 m, then the height of the first tower is

Options

A. `90 m`
B. `(150-60 sqrt(3)) m`
C. `(150 + 20 sqrt(3)) m`
D. None of these

Answer

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

### Step 1: Understand the Problem
We have two towers, with the horizontal distance between them being 60 m. The angle of depression from the top of the second tower to the top of the first tower is 30 degrees. The height of the second tower is given as 150 m. We need to find the height of the first tower.

### Step 2: Draw a Diagram
Visualize the scenario with a diagram. Label the points:
- Let A be the top of the second tower.
- Let B be the top of the first tower.
- Let C be the base of the second tower.
- Let D be the base of the first tower.
- The horizontal distance CD = 60 m.

### Step 3: Identify the Angles
The angle of depression from point A (top of the second tower) to point B (top of the first tower) is 30 degrees. Since angles of depression and elevation are equal when viewed from horizontal lines, the angle CAB (angle of elevation from B to A) is also 30 degrees.

### Step 4: Use Trigonometry
In triangle ABC:
- The height of the second tower (AC) = 150 m.
- The horizontal distance (BC) = 60 m.
- We can use the tangent function since we have the opposite side (AC) and the adjacent side (BC).

Using the tangent of angle CAB:
\[ \tan(30^\circ) = \frac{AC}{BC} \]
Substituting the known values:
\[ \tan(30^\circ) = \frac{150 - h}{60} \]
Where \( h \) is the height of the first tower (BE).

### Step 5: Solve for h
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
So:
\[ \frac{1}{\sqrt{3}} = \frac{150 - h}{60} \]
Cross-multiplying gives:
\[ 60 = \sqrt{3}(150 - h) \]
Expanding this:
\[ 60 = 150\sqrt{3} - h\sqrt{3} \]
Rearranging to solve for \( h \):
\[ h\sqrt{3} = 150\sqrt{3} - 60 \]
\[ h = \frac{150\sqrt{3} - 60}{\sqrt{3}} \]
\[ h = 150 - \frac{60}{\sqrt{3}} \]
Rationalizing the denominator:
\[ h = 150 - 20\sqrt{3} \]

### Step 6: Conclusion
The height of the first tower is:
\[ h = 150 - 20\sqrt{3} \text{ meters} \]