A trough with trapezoidal cross section is formed by turning up the edges of a 30 inch wide sheet of aluminum. So the bottom base I have labeled as 30-2x where each x is the length of the turned up pieces. What I need to do is find the cross section with the maximum amount of area.

trough cross-section

(Remark. This problem can be solved by calculus, although it is somewhat lengthy because you have to maximize with respect to two variables simultaneously (namely the angle to bend and the length of the bent pieces). In this article we will investigate an elementary geometric technique that can be used to solve this and many other problems. If you can't think of a geometrical solution, work it out using calculus, then look at your final answer and think whether there might be a geometrical method to get this.)

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