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Monsieurcoffee, I like your big circle solution. Can you provide a reference to the 'theorem about secants and tangents' on which it is based?
Revising my previous post to remove all reference to transcendental functions yields a solution that is more purely geometric:
Let the near corner of the field be called O, the desired point on the sideline A, the near goalpost B, and the far goalpost C.
Let the angle ACB be called alpha, and the angle BAO be called beta. Note that (in degrees) the desired angle theta = 90 - alpha - beta.
Further note that as x increases, alpha increases and beta decreases.
Conclusion: theta is largest when alpha = beta.
So theta is largest when triangles ACO and BAO are similar.
So theta is largest when x/18 = 8/x.
The solution is x = 12.
__________________
Richard Wallace
Mentor since 2011 for FRC 3620 Average Joes (St. Joseph, Michigan)
Mentor 2002-10 for FRC 931 Perpetual Chaos (St. Louis, Missouri)
 since 2003
I believe in intuition and inspiration. Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.
(Cosmic Religion : With Other Opinions and Aphorisms (1931) by Albert Einstein, p. 97)
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