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Graphing Calculator - Finding Maximum Volume

Updated on December 8, 2011

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 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.


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    • kirbylau profile image

      kirbylau 6 years ago from Wuhan, China

      I concur

    • Storytellersrus profile image

      Barbara 6 years ago from Stepping past clutter

      Any time, Teach!

    • Steve LePoidevin profile image

      Steve LePoidevin 6 years ago from Thailand

      Thanks for stopping by and glad you enjoyed it!

    • Storytellersrus profile image

      Barbara 6 years ago from Stepping past clutter

      Thank God there are people out there who actually keep math alive, lol. I certainly am not a practitioner. I avoid balancing my checkbook if at all possible. I do love to play math games in my head, however. Maybe not the calculator kind, but addition. I like figuring out things without using pad and paper. I suppose this comes from my father, who was pretty brilliant in math.

      Anyway, I think it is fun to read over these hubs and consider practical applications, like volume. This does come in useful. I too often eyeball it, lol. Great job. Voted up and interesting.

    • profile image

      卓峰 6 years ago

      Like! Suggestion: do this problem using Calculus