A vertical tower is 50 m high. Its shadow on level ground is x metres shorter when the Sun's altitude is 45° than when the Sun's altitude is 30°. What is the value of x (the difference in shadow lengths) in metres?

Introduction:
This problem explores how the length of a shadow cast by a vertical tower changes when the Sun's altitude (angle of elevation) changes. We use the tangent function to find the shadow lengths at two angles and then compute the difference between them, which is given as x.

Given Data / Assumptions:

• The tower is vertical and 50 m high.
• Sun’s altitude (angle of elevation) in case 1 = 30°.
• Sun’s altitude in case 2 = 45°.
• The shadow at 45° is x metres shorter than the shadow at 30°.
• Ground is level and there are no obstructions.

Concept / Approach:
For a vertical object of height h and Sun’s altitude θ, the shadow length L satisfies:
tan θ = h / L ⇒ L = h / tan θ.We calculate shadow lengths L₃₀ and L₄₅ for 30° and 45° and then compute x = L₃₀ − L₄₅.

Step-by-Step Solution:
Step 1: Let h = 50 m.Step 2: For θ = 30°, tan 30° = 1/√3.Step 3: Shadow length at 30°: L₃₀ = 50 / tan 30° = 50 / (1/√3) = 50√3.Step 4: For θ = 45°, tan 45° = 1.Step 5: Shadow length at 45°: L₄₅ = 50 / tan 45° = 50 / 1 = 50.Step 6: The shadow at 45° is shorter by x: x = L₃₀ − L₄₅ = 50√3 − 50.Step 7: Factor out 50: x = 50(√3 − 1) metres.

Verification / Alternative check:
Using √3 ≈ 1.732, we get x ≈ 50 * (1.732 − 1) = 50 * 0.732 ≈ 36.6 m. This is a reasonable reduction in shadow length when the Sun’s altitude increases from 30° to 45°. It is also consistent with the general behaviour that shadows become shorter as the Sun appears higher in the sky.

Why Other Options Are Wrong:
50√3 is the longer shadow itself, not the difference. 50 is the shorter shadow length at 45°, not x. 50(√3 + 1) would correspond to a sum, not a difference. 25(√3 − 1) is exactly half of the correct value and does not come from any correct step in the derivation.

Common Pitfalls:
Some students mistakenly compute x as L₄₅ − L₃₀, making it negative, or mix up tan 30° and tan 45°. Others forget to factor the expression, failing to recognise 50(√3 − 1) as the simplified form. Always check which shadow is longer and subtract in the correct order.

Final Answer:
The value of x is 50(√3 − 1) metres.