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Monsieurcoffee, thanks for a very engaging problem.
I changed ISPs this weekend, so my connection was down until now. This gave me some time to think of several solutions.
The obvious one is to express theta in terms of x using the transcendental arctan function, differentiate and solve for the maximum theta.
A little less obvious is to use the pythagorean theorem and the law of sines to express sin(theta) as an algebraic function of x, differentiate and solve for the maximum sin(theta).
More elegant is the following:
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.
Now express alpha and beta as:
alpha = artan(x/18), beta = arctan(8/x)
So theta is largest when x/18 = 8/x.
The solution is x = 12.
The corresponding value of theta is 22.62 degrees.
__________________
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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