For Grandparents' Day in calculus students constructed boxes out of paper. The task: cut squares of equal size out of the corners of an 8.5" x 11" piece of paper and fold up the sides to make a box without a lid. The goal: make a box with the maximum volume. You can see the variety of boxes here.
Next, we used calculus to answer the question. What should the length, x, be for the side of the square to maximize the volume? We created a function and then took the derivative. Students knew that the maximum value would occur where the tangent was horizontal.
I love that this question doesn't lead to an easily factorable quadratic. For a purely calculus and algebraic approach, students actually needed to use the quadratic formula! And there was one solution that we had to throw out. It wasn't the negative solution, because both solutions were positive! Why couldn't x be 4.915? Since the one side of the paper was 8.5 inches, cutting in 4.915 inches from both sides would mean there would be nothing to fold up. The domain for x was 0 < x < 4.25. Finally, we looked at the graph in desmos to confirm our solution.
Who had the box that was closest to being the one with the maximum volume? Caroline S. with squares that were 1.5 inches in length and the volume of her box was 66 cubic inches.
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