Expert: Paul Klarreich - 8/27/2009

Question
Given a 8.5x11 sheet of paper i cut off 2 inches squares out of the corners giving me the dimensions of 7x2x4.5 = 63 cubic inches as my volume. I have found the maximum value as well which is x=1.585417 , y=66.148235. My question is 1.) What size cut(s) will result in a box whose volume is 28? Set up an algebraic equation that could also be used to determine the answer. 2.) Determine the dimensions of a box with exactly 1/2 the maximum volume. Thanks a lot!

Answer
Questioner: Annie
Country: United States
Category: Calculus
Private: Yes <<< Don't do this.
Subject: max/min problem
Question: Given a 8.5x11 sheet of paper i cut off 2 inches squares out of the corners giving me the dimensions of 7x2x4.5 = 63 cubic inches as my volume. I have found the maximum value as well which is x=1.585417 , y=66.148235. My question is 1.) What size cut(s) will result in a box whose volume is 28? Set up an algebraic equation that could also be used to determine the answer. 2.) Determine the dimensions of a box with exactly 1/2 the maximum volume. Thanks a lot!
...................................

It looks like this, I think:  Each 'missing' box is  x-by-x:

        8.5 - x
    +---------------+
    |               |    
    |               |    
+----+---------------+----+
|    |               |    |
|    |               |    |
|    |               |    |
|    |               |    |
|    |               |    |
|    |               |    |
|    |               |    | 11-x
|    |               |    |
|    |               |    |
|    |               |    |
|    |               |    |
|    |               |    |
|    |               |    |
+----+---------------+----+
    |               |    
    |               |    
    +---------------+

When you fold up, your base is 8.5-x  by  11-x, and your height is  x.

Your volume is the product:  

V = x(8.5 - x)(11 - x)

Do your usual min/max stuff to find the max volume.  Then you should be able to handle the rest.