2191 Words5 Pages

The Open Box Problem

An open box is to be made from a sheet of card. Identical squares are

cut off the four corners of the card, as shown below.

[IMAGE]

The card is then folded along the dotted lines to make the box.

The main aim is to determine the size of the square cut which makes

the volume of the box as large as possible for any given rectangular

sheet of card, but first I am going to experiment with a square to

make it easier for me to investigate rectangles.

I am going to begin by investigating a square with a side length of 10

cm. Using this side length, the maximum whole number I can cut off

each corner is 4.9cm, as otherwise I would not have any box left.

I am going to begin by looking into going up in 0.1cm from 0cm being

the cut out of the box corners.

The formula that needs to be used to get the volume of a box is:

Volume = Length * Width * Height

--------------------------------

If I am to use a square of side length 10cm, then I can calculate the

side lengths minus the cut out squares using the following equation.

Volume = Length - (2 * Cut Out) * Width - (2 * Cut Out) * Height

Using a square, both the length & the width are equal. I am using a

length/width of 10cm. I am going to call the cut out "x." Therefore

the equation can be changed to:

Volume = 10 - (2x) * 10 - (2x) * x

If I were using a cut out of length 1cm, the equation for this would

be as follows:

Volume = 10 - (2 * 1) * 10 - *(2 * 1) * 1

So we can work out through this method that the volume of a box with

corners of 1cm² cut out would be:

(10 - 2) * (10 - 2) * 1

8 * 8 * 1

= 64cm³

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