From two points on the same straight line as the base of a vertical tower, at horizontal distances of 4 m and 9 m from the foot of the tower, the angles of elevation of the top of the tower are complementary (their sum is 90°). What is the height of the tower in metres?

Introduction / Context:
This question uses the idea of complementary angles of elevation from two points at different distances from a tower. Complementary angles (adding to 90°) produce a relationship between tangent and cotangent, which we can exploit to find the tower's height without explicitly knowing the individual angles.

Given Data / Assumptions:

• A vertical tower is on level ground.
• Two observation points lie along the same straight line as the tower base.
• Distances from the tower foot: 4 m (nearer point) and 9 m (farther point).
• Angles of elevation of the top of the tower from these two points are complementary (sum = 90°).
• We must find the height of the tower.

Concept / Approach:
If one angle of elevation is θ, the other is 90° − θ. For a height h and horizontal distance d, we have:
tan(θ) = h / d1tan(90° − θ) = cot(θ) = h / d2Because tan(θ) * cot(θ) = 1, we can form an equation involving d1, d2 and h and solve for h.

Step-by-Step Solution:
Let h be the tower height.From the nearer point (distance 4 m): tan(θ) = h / 4.From the farther point (distance 9 m): tan(90° − θ) = cot(θ) = h / 9.We know tan(θ) * cot(θ) = 1.Thus: (h / 4) * (h / 9) = 1.h^2 / 36 = 1 ⇒ h^2 = 36.h = 6 m (positive height only).

Verification / Alternative check:
If h = 6 m, then from d1 = 4 m, tan(θ) = 6 / 4 = 1.5. From d2 = 9 m, tan(90° − θ) = h / 9 = 6 / 9 = 2 / 3. Since 1.5 and 2 / 3 are reciprocals, they are indeed tan(θ) and tan(90° − θ) respectively, confirming complementarity.

Why Other Options Are Wrong:
4 m, 5 m, 7 m, 9 m: None of these values satisfy the product condition (h / 4) * (h / 9) = 1 when squared. They either make the product less than or greater than 1, violating the complementary angle property.

Common Pitfalls:
Students may assume specific angles like 30° and 60° without checking if they satisfy the distance ratios. Another error is adding or subtracting distances and heights instead of using the tangent relationships. Forgetting that tan and cot are reciprocals for complementary angles can also lead to incorrect setups.

Final Answer:
The height of the tower is 6 m.