The shadow of a vertical tower, when the angle of elevation of the sun is 45°, is found to be 10 metres longer than the shadow of the same tower when the angle of elevation of the sun is 60°. What is the height of the tower in metres?

Introduction / Context:
This question connects the height of a vertical tower with the lengths of its shadows at two different angles of elevation of the sun, 45° and 60°. Using right-triangle trigonometry and the relation between tangent, height and shadow length, we can form an equation based on the given 10 m difference in shadow lengths and solve for the tower height.

Given Data / Assumptions:

• The tower stands vertically on level ground.
• Angle of elevation of the sun in case 1 = 45°.
• Angle of elevation of the sun in case 2 = 60°.
• Shadow at 45° is 10 m longer than shadow at 60°.
• We are asked to find the tower height.

Concept / Approach:
For a vertical object of height h casting a shadow of length s, with angle of elevation theta, we have:
tan(theta) = h / s ⇒ s = h / tan(theta)We compute the shadows for 45° and 60° in terms of h and then use the fact that the first shadow is 10 m longer than the second, leading to a simple equation in h.

Step-by-Step Solution:
Let h be the height of the tower.Shadow when angle is 45°: s1 = h / tan(45°) = h / 1 = h.Shadow when angle is 60°: s2 = h / tan(60°) = h / √3.Given: s1 is 10 m longer than s2 ⇒ s1 = s2 + 10.So, h = h / √3 + 10.Rearrange: h - h / √3 = 10.h(1 - 1 / √3) = 10.1 - 1 / √3 = (√3 - 1) / √3, so:h * (√3 - 1) / √3 = 10 ⇒ h = 10√3 / (√3 - 1).Multiply numerator and denominator by (√3 + 1):h = 10√3(√3 + 1) / (3 - 1) = 5√3(√3 + 1).Since √3 * (√3 + 1) = 3 + √3, we get h = 5(3 + √3) m.

Verification / Alternative check:
Approximate √3 ≈ 1.732. Then:
h ≈ 5(3 + 1.732) = 5 * 4.732 ≈ 23.66 m.Shadow at 45°: s1 ≈ 23.66 m.Shadow at 60°: s2 ≈ h / √3 ≈ 23.66 / 1.732 ≈ 13.66 m.Difference s1 − s2 ≈ 10 m, matching the given condition.

Why Other Options Are Wrong:
5(√3 - 1) m and 15(√3 - 1) m: These give much smaller heights and hence smaller shadows, not satisfying a 10 m difference.10(√3 - 1) m and 10(√3 + 1) m: When substituted, the shadow difference is not 10 m; the numbers do not fit the tangent relationships.

Common Pitfalls:
Students sometimes set up the equation as s2 = s1 + 10, reversing the longer and shorter shadow. Others may misuse tan(60°) as 1 / √3 instead of √3, which completely flips the equation. Another common error is failing to simplify fractions involving √3 properly when isolating h.

Final Answer:
The height of the tower is 5(3 + √3) m.