Investigating the Volume of an Open Box

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Investigating the Volume of an Open Box

The aim of this coursework is to investigate the volume of an open box

constructed by one piece of rectangular card that has all four corners

having had squares cut out of them. Firstly I will be studying the

volume whilst changing the side of one length of the cut out square

and the size of the original rectangle card. After I have investigated

this relationship I will try to find out the formula for finding the

cut size to get the largest volume for any specified original card

size.

Square card size

I am going to begin by investigating a square card because this will

give me a basic formula which I can elaborate on. I will start with a

round number of 20cm for the length. This means that the maximum cut

out square length I can cut out will be 9cm else I will have no box

left.

The formula for the volume of any box is as follows:

[IMAGE]

Below, there is a diagram to explain where all these figures come

from.

[IMAGE]

I have also included in this diagram the labels c, x and y, these show

the cut out size and the original length and width of the card, I will

now need to show the values of the width, length and height in terms

ofc, x and y.

[IMAGE]

Therefore I can replace these sub formulae into the first formula.

[IMAGE]

However for the values of a square, y=x therefore:

[IMAGE]

c

V

1

324

2

512

3

588

4

576

5

500

6

384

7

252

8

128

9

36

Unfortunately the value I have chosen for x (20) does not show a very

accurate graph, I will now chose a number of values that show a better

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