Thursday, October 25th

So Mr.Max started off the class by telling us that, outside of math class and teaching, he has probably used applied math to solve problems a total of five times or less. And of those five times, he has used the stuff on maximum volume, so for all of you that were feeling depressed about our little chat last week on how we will probably never use anything we learn in math class outside of school, don't lose hope. :)


We started out by looking at the problem above ▲. Our "building material" was to be 22cm by 28cm, just like a piece of paper. So to make it into a box, we had to cut a square in each corner so the paper would fold nicely into a box. [origami anyone?] To make things easy we started with one cm off each side, but when we made this into a box our class found that the box was quite "shallow". So we thought, what would happen if we made the squares bigger?! While after trying this a couple times we found that the volume of the box seemed to be increasing. But then we kept going with this idea and figured out that, eventually, the volume stops increasing, starts decreasing, and the box begins to get very tall and skinny.



Then Mr.Max showed us that we can make an equation for maximum volume and put it into a spread sheet to generate answers or use our calculators. So to get the equation we started with the formula for volume:

V=(l)(w)(h)

Then we worked out that the formula so that we could plug it into our calculators:

y=(28-2x)(22-2x)(x)

So this basically means that x is equal to the height of the box, and the height of the box is the same as the amount we are taking off each side of the paper, or building material.

First of all Mr.Max showed us how to put it into our calculators. Start by going to Y= and type in the formula. Then hit graph and you should see a cubic function. (you will probably have to play with your window settings a bit) Now to get the maximum volume of this box, go to 2nd Trace, then hit 4. Now it asks for left bound, so move left of the vertex and hit enter. Now do the same for right bound, and then press enter again when it says "guess?".
****x will tell us what the height of the box should be, y will tell us the maximum volume****

If you did it right it will look something like that▼

Then we learned how to do it by spreadsheets. We put all the information into a spreadsheet by making a columns called length, width, height, and volume. So you start off by putting in the dimensions for the first couple boxes, then you can just drag the information down because Excel lets us do that. Then using your spread sheet skills work it out so that in the volume column Excel calculates "length x width x height". So now to find the dimensions of the box with the maximum volume, just scroll down until you come across the highest number in the volume column.

Both methods shown above will produce the same answers for calculating maximum volume. This is proven below: