Question

Consider the following problem: A box with an open top is tobe constructed from a square piece of cardboard, 3 ft wide, bycutting out a square from each of the four corners and bending upthe sides. Find the largest volume that such a box can have. Letx denote the length of the side of the square being cut out.Let y denote the length of the base.

(a)Draw several diagrams to illustrate the situation, some short boxeswith large bases and some tall boxes with small bases. Find thevolumes of several such boxes. (Do this on paper. Your teacher mayask you to turn in this work.)

(b) Draw a diagram illustrating the general situation. Introducenotation and label the diagram with your symbols. (Do this onpaper. Your teacher may ask you to turn in this work.)

(c) Write an expression for the volume V in terms ofx and y.
V =

(d) Use the given information to write an equation that relates thevariables. (Do this on paper. Your teacher may ask you to turn inthis work.)

(e) Use part (d) to write the volume as a function ofx.
V(x) =

(f) Finish solving the problem by finding the largest volume thatsuch a box can have.
V = ft³