• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

The Open Box Problem

Extracts from this document...

Introduction

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.

...read more.

Middle

A = 10

10cm

                                               8cm

                                                       10cm

                                              2cm

                                                            2cm          8cm

Y = A – 2X

The volume of the box = X x Y x Y (length x width x height)

Substitute Y with A – 2X in the equation V = X x Y x Y.

X (A – 2X) ² = V

Check:  

2 (10 – 2x2) ²

2(10 – 4) ²

2 (6) ²

2 x 36

72          

I have produced a graph to show how the volume changes according to the size of the squares cut from each corner of the original. A 10cm x 10cm piece of card was used. 1.1>X>2.2 because I know from previous calculations that the maximum point lies between X = 1cm and X = 2cm

X =  Length of one side of the squares cut out

A = length of one side of the original square before the corners are cut out.

image01.png

This graph shows that the volume reaches its maximum when X = 1.6

I noted down the maximum volume for each size of original card:

...read more.

Conclusion

Before I differentiated I put L in terms of W so the formula is simpler and doesn’t contain too many different letters, which could be confusing.

The following example demonstrates the route to answering the aim. This process can be carried out using any size rectangle but I will be using a rectangle in which L = 2W.

Therefore X (W – 2X) (2W – 2X) will be differentiated with respect to X.

Before differentiation can occur the formula must be expanded then simplified - getting rid of the brackets.

EXPAND:

V = (WX – 2X²) (2W – 2X)

V = 2W²X – 2WX² - 4X²W + 4X³

SIMPLIFY:

2W²X – 6WX² + 4X³

DIFFRENTIATE:

dV/dX must equal 0 for the maximum volume.

dV/dX = 2W² - 12WX + 12X²

This can be simplified by dividing by 2

dV/dX = W² - 6WX + 6X²

I recognized this formula as quadratic so I used the general formula:

To find out what X equaled.

A =  6

B = - 6

C = W

In a rectangle 10cm x 20cm the size of X needed to make the maximum volume is 2.1.

The area of this rectangle is 200cm². 2.1² is cut out from each corner. 2.1² = 4.41

2.205% is cut out from each corner to make the largest possible volume of box.

...read more.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Number Stairs, Grids and Sequences essays

  1. Marked by a teacher

    Mathematics Coursework: problem solving tasks

    3 star(s)

    I will start with putting up tiles in a straight line, then look at two rows of tiles, then three rows of tiles and so fourth. As placing the results into table form proved successful, I will repeat the process again.

  2. What the 'L' - L shape investigation.

    I said before that the number one row up from the L-Number was the grid size and that the number two rows up was double the grid size. Now the L-Shape works in any size grid. As before I stated that the sum of the axis was equal to the last part of the formula.

  1. Open Box Problem.

    the cut of x, which will give the open box its maximum volume, is 6cm. I will now construct a table to prove that this answer is right just like I did for the two squares I investigated before. Cut x L (36-2x)

  2. Investigate the size of the cut out square, from any square sheet of card, ...

    x (2X-2C) x C V= (X-2C) x (2X-2C) C This is my first investigation: 32cm by 16cm piece of card If you look at the table above, you will see that the largest volume, 780cm cubed, was produced by the 3cm by 3cm square cut-out. As I have done in the two previous investigations, I have

  1. Boxes made in the shape of a cube are easy to stack to make ...

    28 As you can see the numbers do not match which means I need to add something else to the formula. I have looked at the numbers and realised that all you need to do is take away N. I then get this: B= 2N2 -N 2N2 - N= 1

  2. My task is to investigate a 2x2 box on a 100 square

    57 58 59 60 57 x 90 = 5130 67 68 69 70 60 x 87 = 5220 77 78 79 80 5220 - 5130 = 90 87 88 89 90 DIFFERENCE = 90 22 23 24 25 22 x 55 = 1210 32 33 34 35 25 x 52

  1. the Open Box Problem

    I would have to prove that this formula/ratio should work on every sized square box. Now I will try a 30X30 square box: Small square length (cm) Volume (cm�) 1 784 2 1352 3 1728 4 1936 5 2000 6 1944 7 1792 8 1568 9 1296 10 1000 11

  2. For my investigation I will be finding out patterns and differences in a number ...

    - 3848 = 40 16 17 18 26 27 28 36 37 38 16 x 38 = 608 36 x 18 = 648 648 - 608 = 40 35 36 37 45 46 47 55 56 57 35 x 57 = 1995 55 x 37 = 2035 2035 - 1995

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work