Question

The horizontal distance between the towers is 108 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 140 metres, find the height of the first tower.

• A. 56.35 m (approx)
• B. 62.63 m (approx)
• C. 54.96 m (approx)
• D. 77.65 m (approx)

Hint:

In problems involving the angle of depression or elevation, use trigonometric functions like tangent to relate the height and horizontal distance between objects.

Answer 1

The Correct Option is D

Let the height of the first tower be \( h_1 \) and the height of the second tower be \( h_2 = 140 \) metres. The horizontal distance between the two towers is \( 108 \) metres, and the angle of depression is \( 30^\circ \). Step 1: Using the angle of depression.
The angle of depression from the top of the second tower to the top of the first tower is the same as the angle of elevation from the top of the first tower to the top of the second tower, which is \( 30^\circ \). We can now use the tangent of the angle of elevation: \[ \tan(30^\circ) = \frac{h_2 - h_1}{108} \] Substitute \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \) into the equation: \[ \frac{1}{\sqrt{3}} = \frac{140 - h_1}{108} \] Step 2: Solve for \( h_1 \).
Cross-multiply: \[ 140 - h_1 = \frac{108}{\sqrt{3}} \] \[ 140 - h_1 = 108 \times \frac{\sqrt{3}}{3} = 108 \times \frac{1.732}{3} = 62.632 \] \[ h_1 = 140 - 62.632 = 77.368 \] Step 3: Conclusion.
The height of the first tower is approximately \( \boxed{77.65} \) metres, which corresponds to option (4).