To find the more precise length of square which I should cut, I have used the upper and lower bounds of the best length of square in the table above (in this case, 2cm).
I have used these results to plot a graph, as shown below. This illustrates the pattern of the results more clearly.
15x15 Square
20x20 Square
Rectangular Boxes
Like what I did with the square cards, I will try cutting different sizes of squares from different sizes of rectangle to find out the square that would create the largest volume.
From looking at the formula, Volume= Length x Width x Height, and inserting the variables I have, I realized I could make a quicker formula to find out the volume. Since finding the length means subtracting two of side length of the square cut from the length of the card, this can be simplified into the formula, Length= a-2x (a=length of card, x=side length of square cut). The same method is applied to the width, Width= b-2x (b=width of the card). Because height of the box is equal to the length of square cut, Height= x.
By substituting these formula’s into the general formula, V=L x W x H, we would be able to calculate the formula just by knowing the length of square cut and the length and width of the sheet of card used, without the trouble of subtracting length and width all the time. The formula would be:
Volume = x (a-2x)(b-2x).
Using this formula, I have calculated the results below.
10x12 Rectangle
10x14 Rectangle
12x14 Rectangle
Analysis
By looking at the graphs, the highest point on the line is the point that reveals the side length of square cut from a box. To find the highest point on the line, I am going to use calculus.
The point shows the volume of the box we want to get. In order to get the volume I used the formula, V = x (a-2x)(b-2x). To get any point in a graph, we need to know the gradient of the line, which is the change the y axis divided by the change in the x axis. Since the point we want is the highest point, we know that the gradient is 0. After expanding and simplifying the formula, we should be able to get a quadratic equation.
=𝟎
Therefore,
Now we want to find x. We can substitute the quadratic equation into the quadratic formula. Then after simplifying the formula, we can get on general formula.
My final general formula
Verification
I substituted the general formula with some results I calculated previously to prove this formula is accurate.
10x12
𝑥
Since it is impossible to two 5.5cm squares off of a 10cm wide card, the answer is 1.8. To make the largest volume for an open box with piece of card 10cm by 12cm, a square of around 1.8cm by 1.8cm is to be cut from each corner. This is verified above in the results table.
20x20
If four 10cm by 10cm squares is be removed from a 20cm by 20cm square card, there will be nothing left. Therefore, 3.3cm is the correct answer which is proven to be true in the results table above.
Conclusion
The general formula I obtained above could be used to find the size of the side length needed to be removed from the sheet of card in order to create an open box a volume as large as possible. I have made sure this formula was accurate trying different sizes of card and all matches my results. While there are many other ways to find the answer, x, I believe by using calculus and quadratics, this formula could give the most accurate and precise answer.