Triangular Boundary Problem

This problem comes from the defunct Wall Street Journal Varsity Math Week collection.

“The coach then shows the team the diagram to the left and asks: What is the maximum area of a rectangle contained entirely within a triangle with sides of 9, 10 and 17?”

I changed the numbers a bit to make my calculations easier, but left the problem otherwise unchanged. When I checked the Varsity Math Week solution, I saw they used a simplifying formula that I could not remember. I also believed their solution left out a justification for the maximal area. Besides an intuitive solution for this, I also included a calculus version. See the Triangular Boundary Problem.

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