This activity connects between patterns and ideas of calculus and can give deep insight and be fun also.
Take a peace of paper (8,5''X11'') and cut the four equal squares from the 4 corners. Fold it to make it into a box w/o top. Keep doing this 4 times and each time cut a square of a different size. Put the boxes side by side and look at the pattern of the volume of the boxes.
Each time you get a different volume isn't it? When the small squares are very tiny the base of the box is big but the hight is small. The volume will then be small, isn't it? On the other side of the spectrum, if the square that you cut is large the height of the box is large but the area of the base becomes quite tiny so again the volume is small. Do you remember by the way the formula of area and volume?
Area= Length times Width and
Volume = Area times Height which is the same as
Volume= Length Times Width Times Height.
So here we are with patterns again, for each square we cut we get a different volume. Draw this pattern on paper, write about it, find a formula...play with it. Now you are hired by a company to make the box with a largest volume. What size square would you cut?
17 comments:
If x is the height (the length of one side of the cut out squares), y is the width, and z is the length, then the equation for the volume of this box is x times y times z. In other words, v=xyz.
Now, to put it in terms of x, y=8.5-2x and z=11-2x. The full equation in terms of x, then, is v=x(8.5-2x)(11-2x). That simplified is v=4x^3-39x^2+93.5x.
Using a calculator, I found the maximum value of v to be roughly 66 when the value for x is about 1.6. Therefore, to maximize the volume of this box, the squares cut out of the piece of paper should be 1.6 inches on each side.
JENNIFER GENOVA
I think that I would make a square that was 1.5 inches per side, which would make the volume the largest possible. The equation goes as follows: 1.5 X 5.5 X 8= 66. I think this is the largest volume possible. I hope I am somewhere close.
I would just like to preface my comment with the fact that this was really fun. This was hands on math, which helped me understand the problem. I think that if more math classes took this approach it would make math less frightening and more enjoyable.
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When I used the 8,5'x11' sheet of paper I started using intervals of 1, then 1'5, 2, and 2'5. I took a picture to show the different perspectives.
1) http://s213.photobucket.com/albums/cc170/ashtron500/WIM/?action=view¤t=001.jpg
2) http://s213.photobucket.com/albums/cc170/ashtron500/WIM/?action=view¤t=003.jpg
3) http://s213.photobucket.com/albums/cc170/ashtron500/WIM/?action=view¤t=005.jpg
The patterns were fun to find and I had a great time stacking up the boxes. Now onto the dimensions of the boxes:
1 INCH BOX: after cutting out the squares, I was left with (6'5x9) which caused the area to be: 58.5in and the volume to be: 58.5in.
1'5 INCH BOX: after I cut out the squares, I was left with (5'5x8) which caused the area to be: 44in and the volume to be: 66in.
2 INCH BOX: after the cutting (4'5x7) the area was: 31.5in and the volume was: 63in.
...now we're getting somewhere.
2'5 INCH BOX: after the cutting (3x6) the area was: 18in and the volume was: 45.
From these calculations I can determine that the largest possible volume is between the (1'5inch) and the (2inch) box. After playing around with the numbers I found that if I made the cuts any smaller or larger than 1'5in that the volume would decrease, therefore I believe that the largest volume can be derived from cutting out the 1'5inch square.
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On another note, I got curious and decided what the volume would look like if I cut 1'5 out of a piece of paper (8'5x8'5). I know that the volume would be smaller but I wanted to know what dimensions I would need to cut in order to make a perfect cubed box.
After fooling around for a bit I kept coming up with the wrong dimensions...could anyone in the class please help me figure this one out?
I found this project to be very interesting, and it also got me thinking about the different approaches to mathematics. I made a lot of attempts with a lot of different size squares, and just by the way of guess and check and "eyeballing" found a similar answer to maximize the volume as i did when i did the equation. I cut out a 1 inch square, a 1.5 inch square, a 2 inch square, and a 3 inch square (which I had started out with and immediately saw was not a good shot at finding the maximum volume!)
For an equation:
y=length
z=width
x=one side of the cut out squares
V=xyz
y=11-2x
z=8.5-2x
(8.5-2x) (11-2x) -i missed foiling!
or (2x-8.5) (2x-11)
=4x^2-39x+93.5
i plugged in a few possible values for x, and found that the largest volume would be 66 for a 1.5 box. I would like to graph this (if i can find my graph paper!) or see it graphed on a calculator to more fully visualize and understand the comparisons between the volumes.
I thought that this was a cool project to do because it was so hands on. It wasnt me just looking at paper and numbers and not caring, I got to do something kinethetic while I figured out the problem.
After messing around with 4 or 5 different size corners I notice that the volume keeps increasing to a certain point, then get smaller again. I didnt have exact numbers when I physically cut the corners out, but the pattern that I saw was pretty clear
The formula that I found that worked was:
v = x(8.5-2x)(11-2x) x= the cut out corner
v= volume
I first plugged in x=1 and got v=58.5. Then I tried the same thing with x=2, this came out to v=63. The volume is stilling increasing so I moved on and made x=3. This time v=37.5, This allowed me to figure out that somewhere between 1 and 2 would be where the volume peaked, because the value of v rose between those two intervals.
Then I plugged in x= 1.1-1.7
x=1.1 v=60.98
x=1.2 v= 62.95
x=1.3 v=64.43
x=1.4 v=65.44
x=1.5 v=66
x=1.6 v=66.14
x=1.7 v=65.89
largest volume = 66.14
x=1.6
I stopped after 1.7 because the volume began to decrease so I knew that anything higher than 1.7 as a value for x would produce a smaller volume than if x=1.6.
In conclusion, I found that if I was hired by a company to produce boxes of this sort with the largest volume. I would cut 1.6 squares out of the corners of the paper.
In class on Friday I determined the formula for the volume of the object to be V=x(8.5-2x)(11-2x)
In reading some of the other responses, I noticed a couple of people getting 1.5 as their answer. I think the problem here was merely that (in Tamar's case) the very first x in the equation was left out.
I remembered in class our looking at the graph of the equation online and the answer being just above 1.5, not 1.5 itself. That said, I believe the maximum volume to be 66 in^3. This value is obtained when 1.6 inches is used for x. Thus, to optimize storage capability, a 1.6 inch square should be cut from each corner.
i did this earlier this morning and TOTAlly forgot to post so hear i go: Although their was some simple math that occurred in the derivation of my answer, it still made it easier to understand the volume concept by actually cutting the paper and seeing it rather than picturing it. The volume of a rectangular prism is V=l*w*h.
BUt when doing the math you need to compensate for the piece you cut out in the paper and can use another variable to reflect that (I'll use Z). SO it will be V=z*l*w.
L= (8.5-2z)
W= (11- 2z)
v=z(8.5-2z)-(11-2z)
After the math you come up with the equation:
1= 58.5
1.3= 64.428
1.6= 66.144
1.9=64.296
2=63
SO the prism with the greatest volume is the one with the 1.6 inch square!
I couldn't find a formula for the largest volume of the paper box. In fact, I didn't make an attempt to really because I became so preoccupied with the idea of cubes. In class I mentioned that it seemed to me that if a square piece of paper was used, cutting out whatever amount would produce a hieght equal to the width and length (making a cube) would have the greatest volume. I think I must've just visualized it this way without thinking about it enough, because I tested it out and the cube is not the biggest. I tried to figure out a formula but I got too frustrated. I'd like to look at it a bit more because I know I'm making it more difficult than it is.
I am in a similar boat as Hannah.
Initially when we did this exercise in class on Friday I thought that each box would have the same volume because each piece of paper was the same size: each time the base got smaller, but the walls got porportionally higher. However, after working with the equations in class I quickly realized that that was wrong.
My next thought was that the piece of paper would need to be a square and a cube with equal sides would create the maximum volume. I tried to play around with the equations but it honestly didn't get me anywhere... I dont think I fully understand this process, and after reading other blogs to try and get ideas I think I just confused myself even more. I would like to go through this in class if possible today to get a better understanding...
I see that my answer was not quite correct-but I did actually multiply the x times the equation that i found- I just accidently left it out in my post (Leaving it out, I assume, would've TOTALLY made the answer off). I realize that my answer was close, but not exact, and that in graphing i would've been able to find the correct answer, 1.6. This has been an interesting example of how even small changes in decimals are important. It is important to be as exact as possible.
I did the same as JENNIFER GENOVA. 1.5 inches per side to get 66. I found that to be the largest volume.
On Friday we found the equation to be V=(x)(8.5-2x)(11-2x)
In class we were just using every whole inch and got:
x y
________
1 58.5
2 63
3 37.5
4 6
Well we can see that the volume is the largest when it is close to 2 in. So i decided to test numbers in between 1 and 2 to see if there was a larger number.
x y
_______
1.1 60
1.2 62.9
1.3 64
1.4 65.4
1.5 66
1.6 66.14
1.7 65.892
So we see that the largest volume was not at 2 inches but was at 1.6 in.
This is great info to know.
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