numerical computation contest

[Christopher149]

Originally Posted by Ether

Close enough. Tell the folks how you did it.

Since this is a circular arc, I start with a circle. Based off of a diagram like this, I draw a triangle and take theta to be as is shown in that picture. From the right triangle, we know radius * sin(theta) = 5280 / 2. From the definition of arc length, we also know that radius * 2 * theta = 5281. We have two equations and two unknowns, so I substituted one into the other and let Mathematica determine theta. From theta, we easily get the radius. Next, we use this equation relating chord length and the height to the radius. Again, I let Mathematica handle the equation to a ridiculous number of decimal places.

Mathematica Code

Clear[t, h, r, c, a]
c = 5280
a = 5281
t = t /. 
  FindRoot[a/(2 t) * Sin[t] == c/2, {t, 1.8}, 
   AccuracyGoal -> 100000, PrecisionGoal -> 100000, 
   WorkingPrecision -> 500]
r = a/(2 t)
FindRoot[r == c^2/(8 h) + h/2, {h, 100}, AccuracyGoal -> 100000, 
 PrecisionGoal -> 100000, WorkingPrecision -> 500]

EDIT: I can get more decimal places ...