Height from angle of elevation at a known horizontal distance From a point on level ground 50 m from the tower base, the angle of elevation to the tower's top is 30°. What is the tower's height?

Introduction / Context:This is a direct tan relation in a right triangle. Angle of elevation θ at horizontal distance d gives height h = d * tan θ. Correctly evaluating tan 30° and simplifying radicals is all that is required.

Given Data / Assumptions:

• Horizontal distance d = 50 m.
• Angle of elevation θ = 30°.
• Level ground, vertical tower.

Concept / Approach:h = d * tan θ = 50 * tan 30° = 50 * (1/√3) = 50/√3 m. If rationalized, h = (50√3)/3 m ≈ 28.87 m. Check options for an exact match in form or value.

Step-by-Step Solution:

Verification / Alternative check:Numerical value: (50√3)/3 ≈ 28.867 m. The provided options list only multiples of √3 like 50√3 m, 75√3 m, etc., none equivalent to 50/√3 m.

Why Other Options Are Wrong:All listed values represent heights far larger than 50/√3 m or in a mismatched form. None equals 50/√3 m.

Common Pitfalls:Using sin instead of tan, or flipping the ratio (taking h/50 = √3 instead of 1/√3). Also note the option formatting: “50 √3 m” is not the same as “50/√3 m.”

Final Answer:None of these