From the top of a hill that is 100 m high, the top and bottom of a nearby vertical tower are seen at angles of depression of 30° and 60° respectively. What is the height of the tower?

Introduction:
This is a trigonometry-based height-and-distance question that uses angles of depression from the top of a hill to the top and bottom of a tower. It tests your understanding of right-angled triangles, angles of depression and the use of tangent functions in solving vertical height problems.

Given Data / Assumptions:

• Height of the hill = 100 m.
• Angle of depression to the top of the tower = 30°.
• Angle of depression to the bottom (base) of the tower = 60°.
• The hill and the tower stand on the same horizontal ground.
• We must find the height of the tower.

Concept / Approach:
Angles of depression from the horizontal at the observer equal the corresponding angles of elevation from the object to the observer. We let the horizontal distance from the hill top to the base of the tower be d, and the height of the tower be H. Using tangent in right-angled triangles:
tan(60°) = vertical drop from hill top to tower base / dtan(30°) = vertical drop from hill top to tower top / d

Step-by-Step Solution:
Step 1: Vertical distances.Hill top is 100 m above ground; tower base is at ground level.Thus, vertical drop to base = 100 m.Vertical drop to top of tower = 100 − H (since tower top is H m above ground).Step 2: Use tan(60°) for base.tan(60°) = 100 / d ⇒ √3 = 100 / d ⇒ d = 100 / √3Step 3: Use tan(30°) for top.tan(30°) = (100 − H) / dBut tan(30°) = 1 / √3 and d = 100 / √3.So, 1 / √3 = (100 − H) / (100 / √3)Step 4: Simplify the equation.1 / √3 = (100 − H) * (√3 / 100)Multiply both sides by √3: 1 = 3 * (100 − H) / 100100 = 3 * (100 − H)100 = 300 − 3H ⇒ 3H = 200 ⇒ H = 200 / 3 ≈ 66.6 m

Verification / Alternative check:
With H ≈ 66.6 m, vertical drop to top is 100 − 66.6 ≈ 33.4 m. Then tan(30°) ≈ 0.577 for 33.4 / d, and tan(60°) ≈ 1.732 for 100 / d. With d = 100 / 1.732 ≈ 57.7 m, both ratios align closely with the standard tangent values, confirming the result.

Why Other Options Are Wrong:
42.2 m, 33.45 m, 58.78 m, 50 m: Substituting any of these heights into the tangent relationships will not satisfy both angle conditions simultaneously. Only H ≈ 66.6 m maintains the correct geometry for both 30° and 60° angles of depression.

Common Pitfalls:
Students may mix up which angle corresponds to the top or bottom, or forget that angles of depression equal angles of elevation. Another frequent error is taking vertical distances incorrectly, for example using H instead of 100 − H for the top of the tower.

Final Answer:
The height of the tower is approximately 66.6 m.