The angle of elevation of the top of a vertical tower from two points on opposite sides of its base, at distances 25 m and 64 m from the foot, are x and 90° − x respectively. Find the height of the tower in metres.

Introduction / Context:
This is a standard height and distance problem in trigonometry. The height of a vertical tower is to be found using angles of elevation measured from two points on opposite sides of the tower. The angles of elevation are complementary, x and 90° − x, and the distances from the base are known. Such problems test understanding of tangent and cotangent relationships in right triangles.

Given Data / Assumptions:

• The tower is vertical and its height is h metres.

• One point is 25 m from the base of the tower and the angle of elevation there is x.

• Another point is 64 m from the base on the opposite side and the angle of elevation there is 90° − x.

• Ground between the points and the tower base is assumed to be horizontal.

Concept / Approach:
For a right triangle with opposite side h and adjacent side d, tan of the angle at the ground is h/d. From the first point, tan x = h/25. From the second point, where the angle is 90° − x, we have tan(90° − x) = h/64. Using the identity tan(90° − x) = cot x = 1 / tan x, we obtain two equations involving h and tan x. Equating the expressions gives a simple relation that allows us to solve for h by eliminating the trigonometric function entirely.

Step-by-Step Solution:
Step 1: From the first point, tan x = h / 25. Step 2: From the second point on the opposite side, tan(90° − x) = h / 64. Step 3: Use the identity tan(90° − x) = cot x = 1 / tan x. Step 4: Therefore h / 64 = 1 / tan x. Step 5: But from Step 1, tan x = h / 25, so 1 / tan x = 25 / h. Step 6: Equate the expressions for tan(90° − x): h / 64 = 25 / h. Step 7: Cross multiply: h^2 = 64 × 25. Step 8: Compute 64 × 25 = 1600, so h^2 = 1600. Step 9: Hence h = √1600 = 40 (taking the positive root since height is positive).

Verification / Alternative check:
We can check consistency by computing tan x and cot x. From h = 40, we have tan x = h/25 = 40/25 = 8/5. Then cot x = 1 / tan x = 5/8. At the second point, h/64 = 40/64 = 5/8, which matches cot x, confirming that tan(90° − x) indeed equals h/64. This confirms that h = 40 m satisfies both angle conditions and distances given in the problem.

Why Other Options Are Wrong:
Option 39 m and 89 m are close in magnitude but do not satisfy h^2 = 64 × 25 and would produce inconsistent values for tan x at the two points. Option 1.6 m is far too small for the given distances and angles. Option 64 m would give tan x = 64/25 and tan(90° − x) = 64/64 = 1, so tan x and cot x would not be reciprocals. Only h = 40 m makes the geometry and trigonometric identities work together correctly.

Common Pitfalls:
A common mistake is to forget that the angles are complementary and to treat tan(90° − x) as tan x instead of cot x. Another pitfall is to misplace the distances and write tan x = 25/h rather than h/25. Some candidates attempt to assign a numerical value to x unnecessarily, even though it cancels out. Focusing directly on tan x and cot x and eliminating the angle leads quickly to the required height.

Final Answer:
The height of the tower is 40 m.