The Open Box Problem

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GCSE Maths Investigation

The Open Box Problem

An open box is made form a sheet of card. Identical squares are then cut from each corner, making a cross shape. The card is then folded to make an open-lid box.

The Yellow squares are the shapes, which are removed. The box is made by folding along the dotted lines.

AIM: The main aim of this investigation is to find the relationship between the size of the rectangle cut and the volume of the box. The size of the rectangle cut which makes the volume of the box as large as possible must be determined. Remembering that a square is also a special form of a rectangle.

As well as the general aim there are two other aims:

  1. For any sized square sheet of card, investigate the size of the cut out square, which makes an open box of the largest volume.
  2. For any sized rectangular sheet of card, investigate the size of the cut out square, which makes an open box of the largest volume.


AIM 1

First I will be looking at aim 1 which uses a square sheet of card.

A square is being cut from each corner. Aim 1 is to find out what fraction of the whole original square needs to be cut from each corner to make the largest possible volume.

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This chart shows the volume compared to the 3 different sizes of original card (10cm², 20cm² and 30cm²) and the size of the squares cut out.

For each size of A (original piece of card) the volume increases to a maximum point decreases there after.

 

Firstly I will find a general rule for the volume in terms of X and A.


To find a general rule for the volume of any box, I will be using the following letters in place of numeric values.

  • X = Length of the side ...

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