An open box is to be made from a sheet of card. Identical squares are cut off the four corners of the card as shown in the diagram below.
10cm
10cm
The card is then folded along the dotted lines to make the box. The main aim of this activity is to determine the size of the square cut out which makes the volume of the box as large as possible for any give square of rectangular piece of card.
In this experiment I will need to investigate at least two squares and two rectangles so I can try and find some formulas that I can use to find out what size square I need to cut out the corner to get the boxes optimum volume. I will start with a 10*10 square and a 12*10 rectangle. I will record all my results in a table like the one shown below.
Then I will go into decimal places when trying to find the optimum volume, then when I have all the results I nee I will plot them onto a line graph.
10*10 Square
1.1=66.924
1.2=69.312
1.3=71.188
1.4=72.576
1.5=73.5
1.6=73.984
1.7=74.052
1.8=73.728
1.9=73.036
Optimum Volume=1 2/3 = 74.074
12*12 Square
1.9=127.756
2.1=127.764
2.2=127.072
2.3=125.948
2.4=124.416
2.5=122.5
2.6=120.224
2.7=117.612
Optimum Volume=2
Mini Summary
I don’t think I have enough results so I’m going to do a 15*15 to increase my results.
15*15 Square
2.1=244.944
2.2=247.192
2.3=248.768
2.4=249.696
2.5=250 Optimum Volume= 2.5 = 250
2.6=249.704
2.7=248.832
2.8=247.408
2.9=245.456
The diagram below will help you to understand my formulas and equations.
10cm (x)
10cm (x)
To find the base area of the square you use the equation (x-2h)(x-2h).
Take h as 1cm
(10-2*1)(10-2*1)=64cm²
To find the size of square to cut out of the corner on a square to find the optimum volume use the formula:- cut-out = Length of side / 6
To find the volume of a square use the formula 100h-40h²+4h³
Take h as 2cm
100*2-40*2²+4*2³
200-160+32 = 232-160
= 72
12*10 Rectangle
1.1=84.084
1.2=87.552
1.3=90.428
1.4=92.736
1.5=94.5
1.6=95.744
1.7=96.492
1.8=96.768
1.9=96.596
Optimum Volume=1.8 = 96.768
15*13 Rectangle
2.1=199.584
2.2=200.552
2.3=200.928
2.4=200.736
2.5=200 Optimum Volume= 2.3 = 200.928
2.6=198.744
2.7=196.992
2.8=194.768
2.9=192.096
30*15 Rectangle Scale: ½
3.1=649.264
3.2=649.472
3.3=648.448
3.4=646.816
3.5=644
3.6=640.224
3.7=635.512
3.8=629.888
3.9=623.376
Optimum Volume=3.2 = 649.472
14*7 Rectangle
1.1=62.304
1.2=64.032
1.3=65.208
1.4=65.856
1.5=66
1.6=65.664
1.7=64.872
1.8=63.648
1.9=62.016
Optimum Volume=1.5 = 66
I have no formula to find the size square needed to be cut out of the corner to make the optimum volume for rectangles such as 12 by 10 or 15 by 13. I have however found a formula that works if one side is double the other. The formula is X/4.7 then that answer /2.
E.g.
30cm
You take the biggest length as X, so in this case it is 30cm
Using the formula the size for the optimum volume is:
30/4.7=6.38
=6.38/2
=3.19cm
If you check my table of results for this size rectangle you will see that I got the optimum volume value as 3.2cm. if I use 3.19 to see what the volume is the volume becomes 649.498, which is the biggest you can get.
If we look at the 14 by 7 rectangle the formula works again.
14cm
So using the formula :-
14/4.7=2.979cm
=2.979/2
=1.489cm
Again check the results for this one and you will see that I found the optimum volume size to be 1.5 with a volume of 66. If I use the size I found with my formula it gives us 66.008 which is slightly bigger.
During this experiment I have found hardly any similarities between the squares and rectangles, but I wasn’t really expecting to. I have however found that as long as one side on a rectangle is double the other, there is a formula that I can use to find out how big I should make the squares that I cut out of the corner to get the optimum volume of the box.
The one similarity I found between the squares and rectangles comes from looking at the graphs I have from the results. Looking at the graphs I can see that they are similar in shape. They all peak in roughly the same area and then begin to fall quite rapidly, this is what I said was what I thought would happen in my prediction.
With the squares I can see from the ones I have looked at that the squares all peak in the high ones and the low twos. Unlike with the rectangles I found a formula I can use with all squares. It tells me what size I should make the squares on the corners for any size square