Quote:
Originally posted by Richard
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.
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That is one solution... however, if you want elegant, check this out:
Draw a circle radius 13 within the rectangular field.
Translate it to the right so that the goalposts lie on the circle.
The goal = 10, the section below = 8
The missing part = x
From a theorum about secants/tangents, the part*(part+whole)=other part*(other part + whole)
so... 8(8+18)=x*x or 8(18)=x^2 which simplifies to 12.