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Find the largest box that can be made from a fixed amount of material

Find the largest open box (i.e., no material covering the top face) that can be made from a rectangular piece of material by removing squares at each corner and turning up the sides in the following cases:

1. the rectangular material has edges 10 and 10;
2. the rectangular material has edges 12 and 18.

1. We begin with a rectangle with sides each of length 10. The picture is: So, the edges of the base of the box are each of length , and the height is ; therefore, the volume is Taking the derivative, Setting this equal to zero and solving, Then looking at the sign of the derivative around these critical points we have Therefore, is increasing on and decreasing on . Hence, has a maximum at . Solving for the length and width of the base of the box we then have 2. This time we begin with a rectangle with length 18 and width 12. The picture is as follows: So, the edges of the base of the box have lengths and , and the height of the box is . Thus, the volume is Taking the derivative, Setting this equal to 0 we obtain the critical points, Then, Hence, has a maximum at . Solving for and we then have 