Height & Distance – Sun’s altitude from shadow ratio A vertical pole casts a shadow on level ground. If the length of the shadow is √3 times the height of the pole, what is the angle of elevation of the Sun (measured from the horizontal)?

Introduction / Context:
Problems that connect a pole’s height and its shadow use right-triangle trigonometry on level ground. The angle of elevation θ of the Sun satisfies tan θ = (opposite)/(adjacent) = height/shadow. Here, the shadow is a known multiple of the height, so the triangle's ratio is fixed.

Given Data / Assumptions:

• Pole is vertical; ground is horizontal.
• Shadow length = √3 × (pole height).
• Angle of elevation θ is acute and measured from the horizontal.

Concept / Approach:
Let height = h and shadow = √3 h. Then tan θ = h/(√3 h) = 1/√3. The standard acute angle whose tangent is 1/√3 is 30°.

Step-by-Step Solution:

Verification / Alternative check:
For θ = 30°, the 30°–60°–90° triangle has opposite:adjacent = 1:√3, matching the given shadow-to-height ratio.

Why Other Options Are Wrong:
60° gives tan = √3, not 1/√3. 90° is vertical Sun (shadow 0). 45° gives tan = 1, which requires equal height and shadow, not √3 times.

Common Pitfalls:
Inverting the ratio (taking shadow/height instead of height/shadow), or confusing 30° and 60° special-angle tangents.

Final Answer:
30°