Maximum Volume of a Box

Date: 04/13/97 at 02:18:47
From: Paul 
Subject: Maximum Volume of a Box

Here's my question.  It has me stumped. Please help and write back 
when you can.

A rectangular sheet of cardboard measures 16cm by 6cm. Equal squares 
are cut out of each corner and the sides are turned up to form an open 
rectangular box. What is the maximum volume of the box?

Thanks,
   Paul

Date: 04/13/97 at 06:44:13
From: Doctor Mitteldorf
Subject: Re: Maximum Volume of a Box

Dear Paul,

Here's the most straightforward way to go about a problem like this:  
Notice that there's just one thing to play with, the size of the 
square.  So let the length of a side of the square corner that's cut 
out be x.  Then the base of the box that you get is 

   (16-2x) * (6-2x)
 
Now write a formula for the volume of the box in terms of x.  Take 
that formula and differentiate it with respect to x.  That shows how
the volume changes when x is increased or decreased.

Set that differentiated formula to equal zero.  This is because if the 
differential were greater than 0, then you could increase x and get 
more volume; if it were less than 0, then you could decrease x and get 
more volume.  Only if it's 0 is there no way to get more volume.  

After you've set the differentiated formula to equal zero, you have an 
equation in x which you can solve.  

Try this, and let me know if it works out for you.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/