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Graphing Calculator - Finding Maximum Volume
Ok, most of the algebra we learn in school isn’t really that useful. Some math teachers probably disagree with this but being a math teacher, I am allowed to say it! I mean really, as I said in my first problem of this little series, how many people really have added x’s and y’s since they left school.
Anyway, this is another little problem that might be of interest to some. Suppose you have a 4’ x 8’ piece of plywood. What is the maximum volume box you can make without a top? With the help of a graphing calculator, we can easily solve this problem.
The answer is found similar to the area problem. Think of unfolding the box into the following pattern.
Volume = length x width x height
= x(8 – 2x)(4 – 2x)
Are we going to do this algebraically? Not on your life! Zip over to http://webgraphing.com/graphing_basic.jsp and plug the formula into the graphing calculator just as it is written above. Click on Graph It and you should get the following graph. I have shaded in the part we want to look at. We don’t care about the negative part because you can’t have a negative length and width!
If you check out the table below, you can see the local maximum (maximum volume in this case) is 12.32 cubic feet for an x value of 0.85. So that means if our box is 0.85 feet by 4 – 2(0.85) feet by 8 -2(0.85) feet, we will have the largest possible volume. You don't have to understand the table to find the numbers!
Therefore, 10” x 27.5” x 75.5” gives us a maximum volume of 12.32 cubic feet, the biggest that is possible from a piece of plywood. Of course, if you are really doing this, you would need to allow a little less here and there for the thickness of the plywood! But this gives you a pretty good idea.
You can extend this same idea to a box with a lid, or a metal cylinder with or without a top and bottom. I’ll write about those two in a later lesson. It is a fairly simple way, with the help of a graphing calculator, to find the maximum volume of any shape you can cut out of a given piece of material.