Revised Formulae
To find the largest volume of an open top box I first need to state that:
This means the cut size needs to be one sixth of the original length.
In the original formula, I stated that, to find the volume of an open top box, where x is one length:
Then if we replace c with x/6:
Multiply out the brackets:
Multiply the brackets by x/6:
Divide by a common denominator:
Simplify:
Finally divide by a factor to simplify:
This means I now have two formulae, one to get the cut size from a value of the length and another using this information to work out the maximum volume without constructing a net or working out the cut size.
Rectangle card size
I am now going to use a rectangle for the original card size; however there are an infinite number of combinations of width and length to try. To combat this I am only using rectangles of the ratio 1:2. I will try to complete at least two examples for this ratio.
Ratio 1:2
Again for the volume of any box I will use the formula:
And again I will need to show the values of the width, length and height in terms of c, x and y yet this time I know that the length is exactly twice the width.
Therefore I can replace these sub formulae into the first formula.
I have constructed a table in excel where I can input the data for the cut size and the original width of the card and it will calculate the volume, I will use it to find the largest volume through the cut size.
x=20
To achieve the most accurate results I managed to go to fourth decimal place where the value for c was 4.2265, this is not however totally accurate as there will be a much larger number of decimals in the c column.
x=30
The value of c for the highest volume is 6.3397 only up to 4 decimal places.
I noticed whilst evaluating the results that if you divide x by c the values are very similar:
This is a very strong relationship; I can therefore predict that if I construct a similar table for another number, if you divide that number by 4.732, it will give its largest volume. I will use number 17: 17/4.732=3.59 therefore I predict that the largest volume will be made with a cut size of approx 3.59.
x=17
The value of c for the highest volume is 3.5925 only up to 4 decimal places.
My prediction proved to be correct and I have added a third row to the data and the average of the three results gives me a formula of the cut size for maximum volume:
This I can include in the formula for area:
Replace c with formula:
Multiply out the brackets:
Multiply by x/4.732071:
Put formula over a common denominator:
Simplify:
This then is the formula for the maximum volume of any open box constructed from a rectangle that’s length is twice as large as its width.
Ratio 1:3
The following formulae shows the values of the width, length and height in terms of c, x and y for the ratio of 1:3 and this time I know that the length is exactly three times the width.
Therefore I can replace these sub formulae into the first formula.
I have constructed a table in excel where I can input the data for the cut size and the original width of the card and it will calculate the volume, I will use it to find the largest volume through the cut size.
x=20
To achieve even more accurate results I managed to go to fifth decimal place where the value for c was, yet even this is not totally accurate as there will be a much larger number of decimals in the c column to get the maximum number for V.
Above is a table that shows the value of x/c. Hopefully this value will be seen again as I complete more values of x, if this happens then I can repeat the steps used to make the formula replacing one value for another.
x=30
The value of c for the highest volume is 6.7712435 only up to 7 decimal places.
Proving my prediction correct, the value of x divided by c the values are very similar:
As before, this is a very strong relationship; I can therefore predict that if I construct a similar table for another number, if you divide that number by 4.4305009, it will give its largest volume. I will use number 17:
17/4.4305009=3.837036 therefore I predict that the largest volume will be made with a cut size of approx 3.837036.
x=17
The value of c for the highest volume is 3.8370379 up to 7 decimal places.
Revised Formulae
My prediction proved to be correct and I have added a third row to the data and the average of the three results gives me a formula of the cut size for maximum volume:
This I can include in the formula for area:
Replace c with formula:
Multiply out the brackets:
Multiply by x/4.732071:
Put formula over a common denominator:
Simplify:
I now have the formulae for both the ratios 1:2 and 1:3, with examples for each.