Answer to Question #2654 in Trigonometry for anand

Question

& & The Statue of Liberty is 46 feet tall, and stands on a platform 47 feet tall. How far from the statue along the ground should I stand to get the largest viewing angle possible? (The "viewing angle" is the angle between the line connecting my eyes to the statue's feet, and the line connecting my eyes to the statue's crown.)

Answer 1

Denote the distance to the Statue as D, the height of it as L2 and the height of the platform as L1. The triangles formed by the heights and the distance is right triangle. Thus the viewing angle would be
alpha = arctan (L₁+L₂)/D - arctan L₁/D = arctan (L₂/D / (1 + L₁(L₁+L₂)/D^₂)) = arctan (L₂D/(D² +L₁(L₁+L₂)) = arctan(46D/(D² + 46(46+47)) = arctan(46D/(D² + 4371))
Let's find the first derivative of this expression and let it to be equal to zero:
arctan(46D/(D² + 4371))' = - 46(D²- 4371)/(D⁴+10,858D² + 19,105,641) = 0
Thus the extrema of this function are D =sqrt(4371) = 66.1 m, D = - sqrt(4371) = -66.1 m (this root doesn't math the conditions or can mean the opposite direction from the statue).
Answer: you should stay 66.11m away from the statue.