Consider the following problem:
a box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. find the largest volume that such a box can have.
1. draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. find the volumes of several such boxes. does it appear that there is a maximum volume? if so, estimate it.