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.)

Hint 1

Draw a picture of the trough, then draw a mirror image reflected across the open top of the trough.

pipe cross-section

This gives you two pieces of aluminum forming a closed pipe. The perimeter of the pipe's cross-section is twice the width of the aluminum sheet, namely 60 inches, and the area is twice the area of the trough's cross-section area. Therefore whatever shape gives the maximum trough size also gives the maximum pipe size, and vice versa. Furthermore the pipe cross-section is a six-sided figure. Now it`s easy to finish: Of all six-sided figures with a given perimeter, which shape has the largest area? Therefore what?

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