[Geometry] Angle Maximization Problem

2021-11-04 09:23 作者:AoiSTZ23 0人读过 | 我要投稿

By: Tao Steven Zheng

【Problem】

This problem was first posed and solved by a late Medieval German astronomer Johannes Müller von Königsberg (1436 – 1476 AD), known more famously as Regiomontanus. The solution presented below uses differentiation. Regiomontanus solved the problem without calculus!

Refer to the above diagram. If the heights $a$ and $b$ are given constants, at what horizontal distance $x$ should one stand to maximize the viewing angle $%5Ctheta$ ? Solve this problem using calculus.

【Solution】

Consider the angles $%5Calpha$ and $%5Cbeta$ such that $%5Ctheta%20%3D%20%5Calpha%20-%20%5Cbeta%20$ . According to the above diagram,

$%5Ctan%7B%5Calpha%7D%20%3D%20%5Cfrac%7Bb%7D%7Bx%7D%20$

$%5Ctan%7B%5Cbeta%7D%20%3D%20%5Cfrac%7Ba%7D%7Bx%7D$

Thus,

$%5Ctheta%20%3D%20%5Carctan%7B%5Cleft(%5Cfrac%7Bb%7D%7Bx%7D%5Cright)%7D%20-%20%5Carctan%7B%5Cleft(%5Cfrac%7Ba%7D%7Bx%7D%5Cright)%7D%20$

To maximize the angle $%5Ctheta$ , derive the above function with respect to $x$ .

$%5Cfrac%7Bd%5Ctheta%7D%7Bdx%7D%20%3D%20%5Cfrac%7B1%7D%7B1%2B%7B%5Cleft(%5Cfrac%7Bb%7D%7Bx%7D%20%5Cright)%7D%5E%7B2%7D%7D%20%5Ccdot%20%5Cfrac%7B-b%7D%7Bx%5E2%7D%20-%20%5Cfrac%7B1%7D%7B1%2B%7B%5Cleft(%5Cfrac%7Ba%7D%7Bx%7D%20%5Cright)%7D%5E%7B2%7D%7D%20%5Ccdot%20%5Cfrac%7B-a%7D%7Bx%5E2%7D%20$

$%5Cfrac%7Bd%5Ctheta%7D%7Bdx%7D%20%3D%20%5Cfrac%7B-b%7D%7Bx%5E2%2Bb%5E2%7D%20%2B%20%5Cfrac%7Ba%7D%7Bx%5E2%2Ba%5E2%7D$

Set $%5Cfrac%7Bd%5Ctheta%7D%7Bdx%7D%20%3D%200$ and solve for $x$ .

$0%20%3D%20%5Cfrac%7B-b%5Cleft(x%5E2%2Ba%5E2%5Cright)%20%2B%20a%5Cleft(x%5E2%2Bb%5E2%5Cright)%7D%7B%5Cleft(x%5E2%2Ba%5E2%5Cright)%20%5Cleft(x%5E2%2Bb%5E2%5Cright)%7D$