The shadow of a tower when the angle of elevation of the Sun is 45° is found to be 10 m longer than when the angle of elevation is 60°. What is the height of the tower?

Introduction / Context:
This problem again uses the relationship between a tower's height and the length of its shadow for two different angles of elevation of the Sun. We are told that when the angle is 45 degrees, the shadow is 10 m longer than when the angle is 60 degrees. The task is to form an equation involving the unknown tower height and then solve it to obtain a simplified expression that matches one of the given options.

Given Data / Assumptions:

• Height of the tower = H (unknown).
• Angle of elevation in first case = 45 degrees.
• Angle of elevation in second case = 60 degrees.
• Shadow at 45 degrees is 10 m longer than shadow at 60 degrees.
• The tower is vertical and stands on level ground.
• Shadow length = height / tan θ for each case.

Concept / Approach:
Let the shadow when the angle is 45 degrees be S1 and when it is 60 degrees be S2. Using tan 45 = 1 and tan 60 = sqrt(3), we can express S1 and S2 in terms of H. The statement that S1 = S2 + 10 allows us to form an equation and solve for H. The algebraic result can then be simplified to an expression involving sqrt(3) that matches one of the choices.

Step-by-Step Solution:
Shadow when angle is 45 degrees: S1 = H / tan 45 = H / 1 = H. Shadow when angle is 60 degrees: S2 = H / tan 60 = H / sqrt(3). Given that S1 is 10 m longer than S2: S1 = S2 + 10. So H = H / sqrt(3) + 10. Rearrange: H − H / sqrt(3) = 10. Factor H: H(1 − 1 / sqrt(3)) = 10. Compute the bracket: 1 − 1 / sqrt(3) = (sqrt(3) − 1) / sqrt(3). Thus H × (sqrt(3) − 1) / sqrt(3) = 10 ⇒ H = 10 sqrt(3) / (sqrt(3) − 1). Multiply numerator and denominator by (sqrt(3) + 1): H = 10 sqrt(3)(sqrt(3) + 1) / (3 − 1) = 5 sqrt(3)(sqrt(3) + 1). So H = 5(3 + sqrt(3)) = 15 + 5√3 m, written as 5√3 + 15 m.

Verification / Alternative check:
Approximate √3 ≈ 1.732. Then H ≈ 5(3 + 1.732) = 5 × 4.732 = 23.66 m. Now S1 = H ≈ 23.66 m and S2 = H / √3 ≈ 23.66 / 1.732 ≈ 13.66 m. The difference S1 − S2 ≈ 23.66 − 13.66 = 10 m, which matches the given condition exactly.

Why Other Options Are Wrong:

• 5(√3 − 1) m and 10(√3 − 1) m: These are much smaller heights and do not produce a 10 m difference between shadows.
• 5(√3 + 1) m: This is again too small; it would give a shorter shadow and a smaller difference.
• 10(√3 + 1) m: This is larger but still does not satisfy the exact equation H − H / √3 = 10.

Common Pitfalls:
Students often write the difference equation in the wrong order, using S2 − S1 = 10 instead of S1 − S2 = 10. Another typical error is to forget to factor out H correctly or to mishandle the rationalisation step involving √3. Carefully keeping track of algebraic manipulations and double checking with an approximate numerical check helps avoid such mistakes.

Final Answer:
The height of the tower is 5√3 + 15 m.