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.

(a) 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.

(b) Draw a diagram illustrating the general situation. Letxdenote the length of the side of the square being cut out. Letydenote the length of the base.

(c) write an expression for the volumeVin terms of bothxandy.

V=

(d) Use the given information to write an equation that relates the variablesxandy.

(e) Use part (d) to write the volume as a function of onlyx.

V(x) =

(f) Finish solving the problem by finding the largest volume that such a box can have.

V=ft³