The length of the shadow of a tower is √3 times the height of the tower. What is the angle of elevation of the Sun?

Introduction / Context:
This problem tests your understanding of how the tangent function relates the height of a vertical object and the length of its shadow. The ratio between the shadow length and the height of the tower is given as √3 to 1. From this ratio, you must infer which standard angle of elevation of the Sun gives tan θ equal to 1 / √3.

Given Data / Assumptions:

• Height of the tower = h (a positive constant).
• Length of the shadow = √3 times the height = √3 h.
• The tower is vertical and the ground is horizontal.
• Angle of elevation of the Sun = θ, where tan θ = opposite / adjacent.
• Opposite side is the tower height and adjacent side is the shadow length.

Concept / Approach:
Using the right triangle formed by the tower, its shadow and the line from the tip of the shadow to the top, we apply tan θ = height / shadow. The given ratio shadow = √3 * height leads to height / shadow = 1 / √3. We match this with known standard trigonometric values: tan 30 degrees = 1 / √3, tan 45 degrees = 1 and tan 60 degrees = √3.

Step-by-Step Solution:
Let the height of the tower be h and the length of its shadow be L. Given L = √3 * h. In the right triangle, tan θ = height / shadow = h / L. Substitute L = √3 h to get tan θ = h / (√3 h) = 1 / √3. We know from standard trigonometry that tan 30 degrees = 1 / √3. Therefore θ = 30 degrees.

Verification / Alternative check:
Recall the basic tan values: tan 30 = 1 / √3, tan 45 = 1, tan 60 = √3. Since we obtained tan θ = 1 / √3, the only compatible acute angle of elevation among the standard choices is 30 degrees. This gives complete agreement with the ratio of shadow length to height provided in the question.

Why Other Options Are Wrong:

• 45°: For 45 degrees, tan 45 = 1, which would require the shadow length and height to be equal, not in the ratio √3 : 1.
• 60°: For 60 degrees, tan 60 = √3, leading to shadow length = height / √3, which contradicts the given condition.
• None of these: There is indeed a matching standard angle, 30 degrees, so this is incorrect.
• 15°: This is not one of the standard angles with tan equal to 1 / √3 and does not satisfy the given ratio.

Common Pitfalls:
Confusing tan θ with shadow / height instead of height / shadow can lead to inverted ratios. Some students also misremember standard trigonometric values, interchanging tan 30 and tan 60. Always verify with the basic right triangle definitions or remember the common exact values for 30, 45 and 60 degrees.

Final Answer:
The angle of elevation of the Sun is 30°.