Open Box Investigation

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25th January


Introduction:

The open box problem is regarding an open box made from a sheet of card. Identical sized squares are removed from the four corners of the card (shown in the diagram below). The aim of this investigation is to verify the size of the square removed that would make the largest volume for any given sheet of card, both square and rectangular. I will do this by finding a general formula which would work for finding the best size of square cut.


Square Boxes

In order to determine the size of square cut off from the card, I tried many different sizes on square boxes to find the size that would give the largest volume. To find the volume I have used the formula:

Volume = Length x Width x Height

To find the length and width I take two of the length of square cut from the total length/width of the box. The height is the same as the side length of square cut.

Below are my results:

10x10 Square

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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 ...

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Here's what a teacher thought of this essay

An excellent piece of work which demonstrates good understanding of volume of cuboids, drawing graphs, use of the formula for solving quadratic equations and substitution into algebraic expressions. Also a good taste of calculus for extension work. Five stars.