Open Box Problem.

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Maths Coursework: Problem Solving Investigation

Tahamtan Pishgharavol

Open Box Problem

Aim

During this project I will be 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.

What is an Open Box

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


Names of things needed for investigation

I will write-up my investigation in Microsoft Word and all formulae shall be calculated on Microsoft Excel and all table and graph will be produce in spreadsheets again in Microsoft Excel.

Structure of investigation

  1. Evidence:  Table
  • Graphs
  • Formulas
  1. Evaluation

1. Evidence

To obtain evidence I will be used a series of methods:

  • Table
  • Graphs
  • Formulae

Part 1, Square

I am going investigate 3 different sizes for the square open box. Once I have obtained all information on the 3 sizes I will look for patterns and try to formulate a rule to work out the largest volume for an open box square. The sizes that I will be using are:

  1. 20 x 20
  2. 40 x 40
  3. 25 x 25

Because it is a square the length = width so, we can write this as L=1W therefore there is a ratio 1:1.

I am going to begin by investigating a square with a side length of 20cm. Using this side length, the maximum whole number I can cut off each corner is 9cm, as otherwise I would not have any box left.

I am going to begin by looking into whole numbers being cut out of the box corners.

 

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

Volume = Length x Width x Height

If I am to use a square of side length 20cm, then I can calculate the side lengths minus the cut out squares using the following equation:

Volume = Length – (2 x Cut Out) x Width – (2 x Cut Out) x Height

OR

Volume = X(L-2X)(L-2X)

= X(L-2X)2

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

Volume = 1(20-2x1)2

So we can work out through this method that the volume of a box with corners of 1cm² cut out would be:

1(20-2x1)2

1 x 18 x 18

= 324cm³

I used these formulae to construct a spreadsheet, which would allow me to quickly and accurately calculate the volume of the box for cut-off size 1cm2 to 9cm2. Below are the results I got through this spreadsheet.

By using a table all results are shown in a clear way and is easier to look for a pattern.

As you can see by the table above, the largest volume is achieved when an area of 3cm² is cut off each corner of the box. I have also drawn a graph to show my results.  By looking at this graph, and my table of results, I can see that to achieve the maximum volume I will need to look at cut outs of between 3 and 4 cm².

  

In the table below I have shown the formulae I have inputted into the spreadsheet to attain the above results.  

To try to make my results more accurate, I am going to investigate the volume of the box with the cut out to more than 1 decimal place. We can see that the maximum box area is made from the cut size of 3cms. Now I will try between 3-4cms

Below is a table showing the cut out to 1 decimal place, with the largest area achieved highlighted in red.

As you can see by the table above, the largest volume is achieved when an area of 3.3cm² is cut off each corner of the box. I have also drawn a graph to show my results. 

Now I’m going to investigate a square with a side length of 40cm. The maximum whole number I can cut off each corner is 19cm, otherwise I would not have any box left. I am going to start off by looking at whole numbers.

Using the formula:

Volume = X(L-2X)2

I used these formulae to construct a spreadsheet, which would allow me to quickly and accurately calculate the volume of the box for cut-off size 1cm2 to 19cm2. Below are the results I got through this spreadsheet.

By using a table all results are shown in a clear way and is easier to look for a pattern.

As you can see by the table above, the largest volume is achieved when an area of 7cm² is cut off each corner of the box. I have also drawn a graph to show my results.  By looking at this graph, and my table of results, I can see that to achieve the maximum volume I will need to look at cut outs of between 6 and 7 cm².

In the table to the left, I have shown the formulae I have inputted into the spreadsheet to attain the above results.  

To try to make my results more accurate, I am going to investigate the volume of the box with the cut out to more than 1 decimal place. We can see that the maximum box area is made from the cut size of 7cms. Now I will try between 6-7cms

Below is a table showing the cut out to 1 decimal place, with the largest area achieved highlighted in red.

As you can see by the table above, the largest volume is achieved when an area of 6.7cm² is cut off each corner of the box. I have also drawn a graph to show my results. 

Now I’m going to investigate a square with a side length of 25cm. The maximum whole number I can cut off each corner is 11cm, otherwise I would not have any box left. I am going to start off by looking at whole numbers.

Using the formula:

Volume = X(L-2X)2

I used these formulae to construct a spreadsheet, which would allow me to quickly and accurately calculate the volume of the box for cut-off size 1cm2 to 111cm2. Below are the results I got through this spreadsheet.

By using a table all results are shown in a clear way and is easier to look for a pattern.

Join now!

As you can see by the table above, the largest volume is achieved when an area of 4cm² is cut off each corner of the box. I have also drawn a graph to show my results.  By looking at this graph, and my table of results, I can see that to achieve the maximum volume I will need to look at cut outs of between 3 and 5 cm².

To try to make my results more accurate, I am going to investigate the volume of the box with the cut out to more than 1 decimal place. ...

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