Length = 6
Breath = 6
Height = 2
Volume = Length x Breath x Height
Volume = 6 x 6 x 2
Volume = 72 cm³
Shape 3
Length = 10 – (Height +Height)
Length = 10 – (3 + 3)
Length = 10 – 6
Length = 4 Length = Breath
Length = 4
Breath = 4
Height = 3
Volume = Length x Breath x Height
Volume = 4 x 4 x 3
Volume = 48 cm³
Shape 4
Length = 10 – (Height + Height)
Length = 10 – (4 + 4)
Length = 10 – 8
Length = 2 Length = Breath
Length = 2
Breath = 2
Height = 4
Volume = Length x Breath x Height
Volume = 2 x 2 x 4
Volume = 16 cm³
From looking at this table, I think I can see a reoccurrence of data forming. Obviously not with the volume of the boxes, but with the fact that the second shape is bigger. I am going to put these figures in another graph, to see if the threat of there being a bigger box exists.
After seeing the graph, I decided that I would continue with searching for a larger volume from my 10 cm by 10 cm square. As shape 2 was again the largest box and I removed a 2 cm by 2 cm square from each corner of it, I will now remove a 1.9 cm by 1.9 cm square from each corner of another 10 by 10 cm square to make shape 5. I will then remove a 2.1 cm by 2.1 cm square from each corner of another 10 by 10 cm square to make shape 6.
I am, as before, predicting that box 2 will remain the biggest.
Shape 5
Length = 10 – (Height + Height)
Length = 10 – (1.9 + 1.9)
Length = 10 – 3.8
Length = 6.2 Length = Breath
Length = 6.2
Breath = 6.2
Height = 1.9
Volume = Length x Breath x Height
Volume = 6.2 x 6.2 x 1.9
Volume = 73.036 cm³
Shape 6
Length = 10 – (Height + Height)
Length = 10 – (2.1 + 2.1)
Length = 10 – 4.2
Length = 5.8 Length = Breath
Length = 5.8
Breath = 5.8
Height = 2.1
Volume = Length x Breath x Height
Volume = 5.8 x 5.8 x 2.1
Volume = 70.644 cm³
It seems that I have predicted wrongly. But this now means that I must delve further into this problem. I now must find out if box 5 is the largest, or if there is a bigger box to be found. So I shall do as before, only this time it will be a 1.5 cm by 1.5 cm square that I will be removing from each corner to make shape 7. Then it will be followed by: -
1.6 cm by 1.6 cm to make shape 8
1.7 cm by 1.7 cm to make shape 9
1.8 cm by 1.8 cm to make shape 10
Shape 7
Length = 10 – (Height + Height)
Length = 10 – (1.5 + 1.5)
Length = 10 – 3
Length = 7 Length = Breath
Length = 7
Breath = 7
Height = 1.5
Volume = Length x Breath x Height
Volume = 7 x 7 x 1.5
Volume = 73.5 cm³
Shape 8
Length = 10 – (Height + Height)
Length = 10 – (1.6 + 1.6)
Length = 10 – 3.2
Length = 6.8 Length = Breath
Length = 6.8
Breath = 6.8
Height = 1.6
Volume = Length x Breath x Height
Volume = 6.8 x 6.8 x 1.6
Volume = 73.984 cm³
Shape 9
Length = 10 – (Height + Height)
Length = 10 – (1.7 + 1.7)
Length = 10 – 3.4
Length = 6.6 Length = Breath
Length = 6.6
Breath = 6.6
Height = 1.7
Volume = Length x Breath x Height
Volume = 6.6 x 6.6 x 1.7
Volume = 74.052 cm³
Shape 10
Length = 10 – (Height + Height)
Length = 10 – (1.8 + 1.8)
Length = 10 – 3.6
Length = 6.4 Length = Breath
Length = 6.4
Breath = 6.4
Height = 1.8
Volume = Length x Breath x Height
Volume = 6.4 x 6.4 x 1.8
Volume = 73.728 cm³
By looking at my results like this, I can see that shape 9 is the largest possible box, you can make from a 10cm by 10cm cube. Now, I want to check my previous formula against the 10 by 10cm cube.
The Formula for Finding the Volume of A Strawberry Box
x x
x x
x x
x x
Length = 10 – 2x
Breath = 10 – 2x
Height = x
Volume = Length x Breath x Height
Volume = (10 – 2x)(10 – 2x) x x
I will now check if my formula is correct
Let x = 3
Volume = (10 – 2x)(10 – 2x) x x
Volume = (10 – 6) x (10 – 6) x 3
Volume = 4 x 4 x 3
Volume = 48 cm³
By comparing my results here with Shape 3 on page 9, I can see that my formula has worked just fine. However, I am going to try it out on yet another size of square (24cm by 24cm square) to be sure that it will work on sizes.
I am going to workout the figures on the next few pages, and place them in a table at the end.
Shape 1
Length = 24 – (Height + Height)
Length = 24 – (1 + 1)
Length = 24 – 2
Length = 22 Length = Breath
Length = 22
Breath = 22
Height = 1
Volume = Length x Breath x Height
Volume = 22 x 22 x 1
Volume = 484 cm³
Shape 2
Length = 24 – (Height + Height)
Length = 24 – (2 + 2)
Length = 24 – 4
Length = 20 Length = Breath
Length = 20
Breath = 20
Height = 2
Volume = Length x Breath x Height
Volume = 20 x 20 x 2
Volume = 800 cm³
Shape 3
Length = 24 – (Height + Height)
Length = 24 – (3 + 3)
Length = 24 – 6
Length = 18 Length = Breath
Length = 18
Breath = 18
Height = 3
Volume = Length x Breath x Height
Volume = 18 x 18 x 3
Volume = 972 cm³
Shape 4
Length = 24 – (Height + Height)
Length = 24 – (4 + 4)
Length = 24 – 8
Length = 16 Length = Breath
Length = 16
Breath = 16
Height = 4
Volume = Length x Breath x Height
Volume = 16 x 16 x 4
Volume = 1024 cm³
Shape 5
Length = 24 – (Height + Height)
Length = 24 – (5 + 5)
Length = 24 – 10
Length = 14 Length = Breath
Length = 14
Breath = 14
Height = 5
Volume = Length x Breath x Height
Volume = 14 x 14 x 5
Volume = 980 cm³
Shape 6
Length = 24 – (Height + Height)
Length = 24 – (6 + 6)
Length = 24 – 12
Length = 12 Length = Breath
Length = 12
Breath = 12
Height = 6
Volume = Length x Breath x Height
Volume = 12 x 12 x 6
Volume = 864 cm³
Shape 7
Length = 24 – (Height + Height)
Length = 24 – (7 + 7)
Length = 24 – 14
Length = 10 Length = Breath
Length = 10
Breath = 10
Height = 7
Volume = Length x Breath x Height
Volume = 10 x 10 x 7
Volume = 700 cm³
Shape 8
Length = 24 – (Height + Height)
Length = 24 – (8 + 8)
Length = 24 – 16
Length = 8 Length = Breath
Length = 8
Breath = 8
Height = 8
Volume = Length x Breath x Height
Volume = 8 x 8 x 8
Volume = 512 cm³
Shape 9
Length = 24 – (Height + Height)
Length = 24 – (9 + 9)
Length = 24 – 18
Length = 6 Length = Breath
Length = 6
Breath = 6
Height = 9
Volume = Length x Breath x Height
Volume = 6 x 6 x 9
Volume = 324 cm³
Shape 10
Length = 24 – (Height + Height)
Length = 24 – (10 + 10)
Length = 24 – 20
Length = 4 Length = Breath
Length = 4
Breath = 4
Height = 10
Volume = Length x Breath x Height
Volume = 4 x 4 x 10
Volume = 160 cm³
Shape 11
Length = 24 – (Height + Height)
Length = 24 – (11 + 11)
Length = 24 – 22
Length = 2 Length = Breath
Length = 2
Breath = 2
Height = 11
Volume = Length x Breath x Height
Volume = 2 x 2 x 11
Volume = 44 cm³
This table is telling me that Shape 4 has the largest volume, but I do not trust that it is the largest box possible, so I will place these results into a graph below.
I am still not convinced that shape 4 is the largest box that I am able to retrieve from a square 24cm by 24cm. So I will double-check this possibility by creating two more boxes. With one, (shape 12) a 3.9cm by 3.9cm square removed from each of the four corners of the 24 by 24cm square. Two, (shape 13) a 4.1cm by 4.1cm square removed from each of the four corners of the 24 by 24cm square.
Shape 12
Length = 24 – (Height + Height)
Length = 24 – (3.9 + 3.9)
Length = 24 – 7.8
Length = 16.2 Length = Breath
Length = 16.2
Breath = 16.2
Height = 3.9
Volume = Length x Breath x Height
Volume = 16.2 x 16.2 x 3.9
Volume = 1023.516 cm³
Shape 13
Length = 24 – (Height + Height)
Length = 24 – (4.1 + 4.1)
Length = 24 – 8.2
Length = 15.8 Length = Breath
Length = 15.8
Breath = 15.8
Height = 4.1
Volume = Length x Breath x Height
Volume = 15.8 x 15.8 x 4.1
Volume = 1023.524 cm³
This table tells me that shape 4 is the largest possible box that you can get from a 24 by 24cm square. But I now have a theory. I am now predicting that my formula will find the volume of any shape of box from any size of square.
Now, because I have made a statement, I must back it up with some kind of evidence, but because I have already proven that it will work with two other size squares, I now have to prove that it will work with the 24cm by 24cm square.
The Formula for Finding the Volume of A Strawberry Box
x x
x x
x x
x x
Length = 24 – 2x
Breath = 24 – 2x
Height = x
Volume = Length x Breath x Height
Volume = (24 – 2x)(24 – 2x) x x
I will now check if my formula is correct
Let x = 3
Volume = (24 – 2x)(24 – 2x) x x
Volume = (24 – 6) x (24 – 6) x 3
Volume = 18 x 18 x 3
Volume = 972 cm³
After calculating my formula, I have once again realised that it was correct. And because the formula worked yet again, it has proven that my formula can and will work for any shape of box made from any size of square.
Now that I have proven that the formula works for all three squares, I must now see if there is a connection between the size of the squares and the box with the maximum volume.
I see straight away that there is a connection between the 12 by 12 cm square and the 24 by 24 cm square. I can see that the size of the square removed to give the largest volume is exactly one 6th of one size of the original shape. This can also be written as 1/6. Which means, when I divide the length or the breath by 6, I get the size of square that has to be removed to get the largest volume. I will now check this against the 10 by 10 cm square.
10/6 = 1 4/6 = 1 2/3 2/3 as a decimal = 0.7
so 1 2/3 = 1.7 When this is tried with the other squares, you can see that this works for all three squares.
As there is a formula for finding out the size of corner to remove to give you the largest volume, I believe that there has to be a formula for finding the largest volume without doing three or four different calculations. So I shall be continuing this investigation further by trying to find this formula.
Make any side of square equal x, and any side of box equal y.
If one side of the square removed from corner of original square equals 1/6 of x, then x must equal 6/6.
As, to make the box, we must remove a square from each corner of the original square, it means that 1/6 is being taken away from x twice. This then means that 2/6 is being taken away from x to get y.
So y= 6/6 – 2/6 = 4/6 = 2/3
y= 2/3
Because this is a square, then y equals the length of the side of the box, but y also equals the breadth of the box. This means that the length of the box (L) equals 2/3x, the breadth of the box (B) equals 2/3x and because the size of one side of the square removed from each corner is the height of the box, height (H) equals 1/6x.
So, as volume = L X B X H… our new formula is
2/3x X 2/3x X 1/6x
I will now test this formula, and for which, x will equal 24. I am predicting a finding that will prove that this formula works.
Volume= 2/3x X 2/3x X 1/6x
2/3 of 24 = 24 ÷ 3 X 2 1/6 of 24 = 24 ÷ 6
= 8 X 2 = 4
= 16
2/3x X 2/3x X 1/6x
= 16 X 16 X 4
= 1024 cm³
So by looking back to the table on page 23, I can see that this formula works for finding the maximum volume of a strawberry box with a size of 24 by 24cm. Although before I am completely convinced that this will work, I am going to try it out again, this time x will equal 12.
Volume= 2/3x X 2/3x X 1/6x
2/3 of 12 = 12 ÷ 3 X 2 1/6 of 12 = 12 ÷ 6
= 4 X 2 = 2
= 8
2/3x X 2/3x X 1/6x
= 8 X 8 X 2
= 128 cm³
And now, by looking back to the table on page 6, I can clearly see that this formula has worked, and from this I can say that this formula will work to find the maximum volume of a strawberry box using any size of square. But there has to be a smaller and quicker formula than this one. So to try and find it, I am going to multiply the formula together.
2/3x X 2/3x X 1/6x = 4/5x³
= 2/27x³
I now must check if this formula is correct, and for this, I will use the 12 by 12cm square.
2/27x³ = 2/27 X 12³ = 2/27 X 1728
= 2 ÷ 27 X 1728 = 128cm³
And by looking back again to the table on page 6, I can see that my new formula, 2/27x³, works to find the largest volume possible for a 12 by 12cm square, but I want to check it with the 24 by 24 cm square. I am predicting that it will work, but I need to be certain.
2/27x³ = 2/27 X 24³ = 2/27 X 13824
= 2 ÷ 27 X 13824 = 1024cm³
And by looking back to the table on page 23, I can see that the formula will work to find the largest possible volume for any square, but what about other shapes? Would it work using a rectangle as well, or would I need a different formula?
The answers to these questions I am determined to find out. So I will now try out the same method with rectangles. But because rectangles can have any ratio of length by breadth, I am going to keep all of the rectangle shapes to a ratio of 1:2 starting with 12 by 24cm, then 15 by 30cm and finally, 24 by 48cm.
To solve this problem I will have to work out the same way I did with the squares. Which was, by taking a rectangle piece of paper, cutting out smaller squares from each corner, and folding up the sides to make a box shape. Then repeating this until, I can find the largest possible volume for each size of rectangle. Below is an image that illustrates the 12 by 24cm rectangle to show you what I had to work with.
(Please note: All diagrams are not drawn to scale, and this will be the last diagram)
I will now proceed to find the maximum volume possible for the 12 by 24cm rectangle as I did before with the squares, only this time I will work-out the volume using the formula that is on pages 7, 14 and 24 because I believe that will take up less space.
Shape 1
Size of square removed = x = 1
Length = 24 – 2x
Breadth = 12 – 2x
Height = x
Volume = (24 – 2x)(12 – 2x) X x
Vol = (24 – 2)(12 – 2) X 1
Vol = 22 X 10 X 1
Vol = 220cm³
Shape 2
Size of square removed = x = 2
Length = 24 – 2x
Breadth = 12 – 2x
Height = x
Volume = (24 – 2x)(12 – 2x) X x
Vol = (24 – 4)(12 – 4) X 2
Vol = 20 X 8 X 2
Vol = 320cm³
Shape 3
Size of square removed = x = 3
Length = 24 – 2x
Breadth = 12 – 2x
Height = x
Volume = (24 – 2x)(12 – 2x) X x
Vol = (24 – 6)(12 – 6) X 3
Vol = 18 X 6 X 3
Vol = 324cm³
Shape 4
Size of square removed = x = 4
Length = 24 – 2x
Breadth = 12 – 2x
Height = x
Volume = (24 – 2x)(12 – 2x) X x
Vol = (24 – 8)(12 – 8) X 4
Vol = 16 X 4 X 4
Vol = 256cm³
Shape 5
Size of square removed = x = 5
Length = 24 – 2x
Breadth = 12 – 2x
Height = x
Volume = (24 – 2x)(12 – 2x) X x
Vol = (24 – 10)(12 – 10) X 5
Vol = 14 X 2 X 5
Vol = 140cm³
As before, I will place the results for finding the volume of a rectangular box with a net of size 12 by 24cm, in a table displayed below.
To make this information more presentable, I am going to put it into a graph below.
I have learned from working out the largest volume using the square, that the first table/graph doesn’t always show that largest volume of box. This table and graph shows that the largest box has been found by removing a 2cm square from each corner of the original rectangle. I am now going to check weather or not Shape 2 is the largest shape. Checking this will be done by removing a 1.9cm square from each corner of the original rectangle to make Shape 6, and removing a 2.1cm square from each corner of the original rectangle to make Shape 7.
Shape 6
Size of square removed = x = 1.9
Length = 24 – 2x
Breadth = 12 – 2x
Height = x
Volume = (24 – 2x)(12 – 2x) X x
Vol = (24 – 3.8)(12 – 3.8) X 1.9
Vol = 20.2 X 8.2 X 1.9
Vol = 314.716cm³
Shape 7
Size of square removed = x = 2.1
Length = 24 – 2x
Breadth = 12 – 2x
Height = x
Volume = (24 – 2x)(12 – 2x) X x
Vol = (24 – 4.2)(12 – 4.2) X 2.1
Vol = 19.8 X 7.8 X 2.1
Vol = 324.324cm³
I am now going to place these two new figures in a table along with Shape 2 so that I can compare my new results with my last ones.
Barry McNickle
12 E
