Expert: Paul Klarreich - 1/21/2007

Question
A box with a closed top is to be made from a 6-ft by 10-ft piece of cardboard by cutting out four squares of equal size, folding along the dashed lines, and tucking the two extra flaps inside.
a. Find a formula that expressed the volume of the box as a function of the length of the sides of the cut-out squares.
b. Find an inequality that specifies the domain of the function in part a.
c. Estimate the dimensions of the box of largest volume.

Answer
Questioner:   Amanda
Category:  Calculus
 
Subject:  Optimization
Question:  A box with a closed top is to be made from a 6-ft by 10-ft piece of cardboard by cutting out four squares of equal size, folding along the dashed lines, and tucking the two extra flaps inside.
a. Find a formula that expresses the volume of the box as a function of the length of the sides of the cut-out squares.
b. Find an inequality that specifies the domain of the function in part a.
c. Estimate the dimensions of the box of largest volume.

.....................................
Hi, Amanda,

This is one of your 'classic' max-min problems.  For these, you have to use the spatial-relations section of your brain (right behind the hippocampus, I think) and imagine that you are making the box like this:

WARNING: THE FOLLOWING DISCUSSION MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.

Your sheet of cardboard:

<----------- 10 feet  -------->
+-----------------------------+
|                             |
|                             |
|                             |
|                             | 6 feet
|                             |
|                             |
|                             |
+-----------------------------+

becomes, after you make the cutouts:

   <----- 10 - 2x ------->
   +---------------------+
   |x                   x|   
+---+                     +---|
| x                         x |
|                             | 6 - 2x
| x                         x |
+---+                     +---|
   |x                  x |   
   +---------------------+

Now that part of your brain tells you that after you fold up the flaps, you have a box whose

Height  is x
Width   is 6 - 2x
Length  is 10 - 2x

and whose volume is  LWH = x(6 - 2x)(10 - 2x)

Then you will conclude that the value of x, the size of the cutout, must be >= 0, and also must be  <= 3, otherwise the Width. 6-2x, would be negative.

Now you can do better than estimate the maximum volume; you can determine it:

A. Let  V = x(6 - 2x)(10 - 2x)

B. Differentiate that (Multiply out, of course, before you try.)  and set the derivative equal to zero.

C. Solve that equation for x.

D. Substitute  x = (answer to C),  x = 0,  and  x = 3, which are the endpoints of your domain interval.  Whichever gives you the largest V is your answer.  I'm betting on the answer to C.