A tower is 50 m high. Its shadow is x metres shorter when the Sun's altitude is 45° than when it is 30°. What is the value of x in metres?

Introduction / Context:
This tower and shadow problem focuses on how the length of a shadow changes when the Sun's altitude (angle of elevation) changes. You are given the fixed height of the tower and told that the shadow is x metres shorter at an altitude of 45 degrees than at 30 degrees. The goal is to express this difference in shadow lengths in terms of standard trigonometric values involving sqrt(3).

Given Data / Assumptions:

• Height of the tower, H = 50 m.
• Angle of elevation of the Sun in the first case = 30 degrees.
• Angle of elevation of the Sun in the second case = 45 degrees.
• The shadow at 45 degrees is x metres shorter than the shadow at 30 degrees.
• Level ground and a perfectly vertical tower are assumed.
• Standard values: tan 30 = 1 / sqrt(3), tan 45 = 1.

Concept / Approach:
The length of the shadow is given by shadow = height / tan θ. We compute the shadows at 30 degrees and 45 degrees separately and then take the difference. Since the shadow becomes shorter when the Sun is higher (greater angle), the longer shadow will correspond to 30 degrees. The difference (longer minus shorter) is given as x and must be expressed in a factorised form involving sqrt(3).

Step-by-Step Solution:
Let shadow_30 be the shadow when the angle is 30 degrees. shadow_30 = H / tan 30 = 50 / (1 / sqrt(3)) = 50 * sqrt(3) m. Let shadow_45 be the shadow when the angle is 45 degrees. shadow_45 = H / tan 45 = 50 / 1 = 50 m. Given that shadow_30 - shadow_45 = x. So x = 50√3 - 50 = 50(√3 - 1) m.

Verification / Alternative check:
We know tan 30 is smaller than tan 45, so shadow_30 must be longer than shadow_45, which we see clearly: approximately 50 * 1.732 = 86.6 m versus 50 m. The difference 86.6 - 50 = 36.6 m, which is consistent with 50(√3 - 1) since (√3 - 1) is about 0.732. This confirms that the algebraic form matches the approximate numeric value.

Why Other Options Are Wrong:

• 50√3 m: This equals the shadow at 30 degrees alone, not the difference.
• 50(√3 + 1) m: This value is too large and does not represent a simple difference between the two shadows.
• 50 m: This equals the entire shorter shadow, not the difference of two shadows.
• 25(√3 - 1) m: This is exactly half of the correct difference and arises if someone accidentally divides by 2.

Common Pitfalls:
Students may reverse the shadows and incorrectly assume that the shadow at 45 degrees is longer. Another mistake is to use the formula shadow = height * tan θ instead of height / tan θ. Additionally, algebraic factorisation errors, such as not recognising 50√3 - 50 as 50(√3 - 1), can cause confusion when matching with given options.

Final Answer:
The value of x, the amount by which the shadow is shorter at 45 degrees than at 30 degrees, is 50(√3 - 1) m.