Solution to Problem 77

Congratulations to this week's winners

Paul Leisher, Nathan Pauli

Correct solutions were also submitted by Yan Fridman, Bill Webb, Cyril Terakopiantz, Ron Welch, Emanuele Macri, Loranzo Pozzoli, Parsa.

Label the legs of the triangle as in the diagram on the above. The law of cosines and Pythagoras theorem gives

18.5² = s₁²+ s₂² - 2 s₁s₂cos q,
s₁(x) = Ö(59.25² + 10² + x²),
s₂(x) = Ö(40.75² + 10² + x²).

(Notice that s₁ and s₂ are functions of x). Therefore

A quick sketch of the graph of q as a function of x shows that q is increasing for x less than about 48.10 and decreasing after that. Therefore the optimal angle occurs at a distance of a little more than 15 yards from the goal post.

For the post 1974 rules, the above analysis, appropriately modified, will show that the angle q is always decreasing for positive values of x.

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Page last updated 13 January 2000.