(length-2χ)² x χ
(χ is the cut out square length)
Because the size of the square sheet of card is 18cm², the maximum length the cut out square can be is 8cm by 8cm because if I made the cut out square 9 by 9 the whole square would just be separated into four quarters and there would be no base left. Therefore I am going to do 6 trials going up in 1cm intervals starting from 2cm up to 8cm cut lengths.
Below is the table of results I obtained for the 18cm by 18cm by using the volume formula.
Square paper 18 x 18cm
As you can see from the table the largest volume of the open box was with the cut of 3cm. This shows that the cut out square that gives the 18cm by 18cm box its largest volume is 3cm. However this may not be correct as the size of the cut is rounded to the nearest whole number so the cut that gives the largest volume may be between 2.5cm and 3.5cm. Below I have calculated the volumes of square cuts between 2.5 and 3.5 going up in 0.1 intervals.
This shows clearly that the cut out square with a length of 3cm when folded along the cuts gives the largest possible volume for 18cm² square sheets. I believe this to be an accurate enough measurement so I will not go into 2 decimal places.
By looking at the results I have discovered a relationship between the length of the square and the height (cut out square). I calculated that 3cm is 1/6 of 20cm. This shows that a cut that is 1/6 of the length of the square sheet produces an open box of the largest volume. However this rule may not apply to all square sheets, so I will have to investigate two more square sheets to see if this rule applies to all square sheets. The next square sheet I am going to investigate is 24cm by 24cm.
I will do 10 trials because if I go any higher the square will be divided into four separate quarters. I will be going up in 1cm intervals in a range of 2-11cm cuts, starting from 2cm.
Square paper 24 x 24
As you can see from the table the cut which gives the largest volume of the open box is 4cm. This shows that the cut out square which gives the 24 by 24cm box its largest volume is 4cm. As I have done before with the 18 by 18 box I will find the cut out square which gives the largest volume by finding the volumes of cut out squares between 3.5 and 4.5 to give the exact size of the cut
This shows clearly that a cut out square with a length of 4cm when folded along the cuts gives the largest possible volume for 24cm² sheets. I believe this to be an accurate enough measurement so I will not go into 2 decimal places.
The relationship in these results is similar to the relationship in the 18 by 18cm square. The results show that the height (4cm) is 1/6 of the length of the square sheet. This again proves the rule that a cut which is 1/6 of the length of the square produces an open box with the largest volume. To make absolutely sure that this rule applies to all square sheets I will investigate one more square which is 30 by 30cm. For this square I will do 7 trials because if I go any higher the square will be split up into four quarters. I will go up in 1cm intervals in a range of 2-30 cut lengths starting from 2cm.
Square paper 30 x 30
As you can see from the results the cut out square which gives the largest volume of the open box is 5cm. This shows the cut out square which gives the 30 by 30cm square its largest volume is 4cm. This may not be the correct size of the cut because it is round to the nearest whole measurement therefore I will do the same as before and find the volumes for the cut length between 4.5 and 5.5cm.
This clearly shows clearly that a cut out square with a length of 5cm when folded along the cuts produces an open box of the largest volume for a 30cm² sheets. I believe this to be an accurate enough measurement so I will not go into 2 decimal places.
The relationship in these results is the same as the previous two sets of results. The results show that the height (cut out square) which is 5cm is 1/6 of the length of the square sheet which is 30cm. This definitely proves the rule that says a cut which is 1/6 of the whole length of the square sheet produces an open box with the largest volume.
