I have found the following results:
Grid Size: 12cm x 12cm
Grid Size: 18cm x 18cm
Grid Size: 24cm x 24cm
Shaded numbers are ones that give the maximum volume.
With the square that has a grid size of 12cm x 12cm the cut size that gives the maximum volume is 2cm and I have tested this by trying cut sizes slightly smaller and slightly larger than the cut size that appears to give the maximum volume.
The grid size of 18cm x 18cm has a maximum volume which comes from the cut size of 3cm and again this has been tested by trying cut sizes slightly smaller and slightly larger than the cut size that appears to give the maximum volume.
With a grid size of 24cm x 24cm the cut size of 4cm gives the maximum volume and this has been tested again by trying cut sizes slightly smaller and slightly larger than the cut size that appears to give the maximum volume.
It seems that by dividing the grid size by 6 you can get the cut size that gives the maximum volume. For example a 1/6 of 12 equals 2, 1/6 of 18 equals 3 and 1/6 of 24 equals 4.
The length of the box can be generalised as y-2x
Where: y = length
x = cut size length
v = volume
x
y
Using this diagram I have formed an equation for the volume of the box. First you multiply the length by the width giving you the formula (y-2x)(y-2x). Then you multiply that by the height to get v= x((y-2x)(y-2x)) which cancels down to v= x(y-x)2
The Open Box Problem: Part 2
In this section I will be investigating the largest volume within any sized rectangle and the link it has with its square cut out.
The volume of the cuboids will be found by multiplying the length by the width by the height for several different sized rectangles until a pattern is found.
To find the maximum volume I will increase the size of the square cut out until I appear to reach a maximum. Then I will increase and decrease the size of the cut out slightly to check that I have found the maximum volume.
For my first results the ratio of length to width is 2:1. Note all extra figures in the graphs and tables have been added to give better quality results on the graphs. Again as with the square the volume of the rectangle increases up to a certain cut size then decreases as you increase the size of the cut size further.
For three rectangles I have done where the length is twice the width I have the following results:
For the 20:10 the cut size of 2.1cm gave the maximum volume, for the 40:20 it was a cut size of 4.2cm and for the 80:40 a cut size of 8.5cm gave the maximum volume. Each of these cut sizes are approximately 1/5 of their original box width which is quite different to the square where you could find the exact maximum volume by calculating 1/6 of the grid size.
x = length
y = width
z = cut size
z = height
Length of open box = x-2z
Width of open box = y-2z
The cut size that gives the maximum volume is about 1/5 of the width so this can be written as y/5. For the exact maximum volume I have made a formula which is z((y-2z)(x-2z)).
I have also investigated other ratios of, 1:3, 1:4 and 1:5 and have discovered that that the ratio of length to width doesn’t make a difference to how the maximum volume is found and the same rules as before apply to these rectangles as well.
Ratios of 2:1
Ratios of 3:1, 4:1 and 5:1
