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Open Box Problem Coursework Columbia

The Open Box Problem

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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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Related Searches

Open Box Problem         Main Aim         Width         Corners         Spreadsheet         Volume         Maximum         Equation        





I used these formulae to construct a spreadsheet, which would allow me
to quickly and accurately calculate the volume of the box. Below are
the results I got through this spreadsheet:

CARD SIZE: 10CM BY 10CM

LEGNTH AND WIDTH OF THE CUT OUT SQUARE

LEGNTH

WIDTH

HEIGHT

VOLUME

0.00

10.00

10.00

0.00

0.000

0.10

9.80

9.80

0.10

9.604

0.20

9.60

9.60

0.20

18.432

0.30

9.40

9.40

0.30

26.508

0.40

9.20

9.20

0.40

33.856

0.50

9.00

9.00

0.50

40.500

0.60

8.80

8.80

0.60

46.464

0.70

8.60

8.60

0.70

51.772

0.80

8.40

8.40

0.80

56.448

0.90

8.20

8.20

0.90

60.516

1.00

8.00

8.00

1.00

64.000

1.10

7.80

7.80

1.10

66.924

1.20

7.60

7.60

1.20

69.312

1.30

7.40

7.40

1.30

71.188

1.40

7.20

7.20

1.40

72.576

1.50

7.00

7.00

1.50

73.500

1.60

6.80

6.80

1.60

73.984

1.70

6.60

6.60

1.70

74.052

1.80

6.40

6.40

1.80

73.728

1.90

6.20

6.20

1.90

73.036

2.00

6.00

6.00

2.00

72.000

As you can see the cut-out size which gives the largest volume is 1.70
cm but to find a more accurate cut-out size I am to find the maximum
cut-out size between 1.60cm and 1.80cm:

CARD SIZE: 10CM BY 10CM- TO FIND EXACT CUT-OUT SIZE

LEGNTH AND WIDTH OF THE CUT-OUT SIZE

LEGNTH

WIDTH

HEIGHT

VOLUME

1.63

6.74

6.74

1.63

74.047

1.64

6.72

6.72

1.64

74.060

1.65

6.70

6.70

1.65

74.069

1.66

6.68

6.68

1.66

74.073

1.67

6.66

6.66

1.67

74.074

1.68

6.64

6.64

1.68

74.071

1.69

6.62

6.62

1.69

74.063

1.7

6.6

6.6

1.7

74.052

As you can see by the table above, the largest volume is achieved when
the cut-out size of each corner of the box is 1.67cm. I also made a
graph to prove that the maximum cut-out size is around 1.67cm.

[IMAGE]

As you can see the graph shows that the maximum cut out size is
between 1.5cm and 1.7cm.

If I wish to work out the proportion of the box that needs to be cut
away to obtain the maximum cut-out size, I need to divide 1.67cm by
10. In doing this I get an answer of 0.16665, or a proportion of 1/6.
To see if this is correct I decided to look at different size square,
15cm by 15cm. I worked out a 1/6 of 15, which was 2.5cm so I made a
spreadsheet to work out the maximum cut out size and volume of a 15cm
by 15cm square. I looked between 2.41cm and 2.51cm for the maximum cut
out size, as I knew the cut out size should be around that figure.

15cm by 15cm square

LEGNTH AND WIDTH OF THE CUT OUT SQUARE

LEGNTH

WIDTH

HEIGHT

VOLUME

2.41

10.18

10.18

2.41

249.754

2.42

10.16

10.16

2.42

249.806

2.43

10.14

10.14

2.43

249.852

2.44

10.12

10.12

2.44

249.891

2.45

10.10

10.10

2.45

249.925

2.46

10.08

10.08

2.46

249.952

2.47

10.06

10.06

2.47

249.973

2.48

10.04

10.04

2.48

249.988

2.49

10.02

10.02

2.49

249.997

2.50

10.00

10.00

2.50

250.000

2.51

9.98

9.98

2.51

249.997

As you can see from the table above the proportion of the box that
needs to be cut away to obtain the maximum cut out size is 1/6 as I
found out the maximum cut-out size is 2.5cm. I also made a graph to
prove that 2.5cm is the maximum cut out size.

[IMAGE]

To prove that 1/6 of the size of the box gets the maximum cut out size
again, I tried to prove it with a box with measurements of 20cm by
20cm. So I worked out 1/6 of 20 which gave me 3.333 so in the
spreadsheet I used numbers between 3 and 3.6 for the cut out square.

LEGNTH AND WIDTH OF THE CUT-OUT SIZE

LEGNTH

WIDTH

HEIGHT

VOLUME

3.00

14.00

14.00

3.00

588.000

3.10

13.80

13.80

3.10

590.364

3.20

13.60

13.60

3.20

591.872

3.30

13.40

13.40

3.30

592.548

3.40

13.20

13.20

3.40

592.416

3.50

13.00

13.00

3.50

591.500

3.60

12.80

12.80

3.60

589.824

As you can see the maximum cut out size is 3.30, which is around the
answer, I worked out. So this definitely proves that 1/6 of the length
or width of the box gives me the maximum cut out size of the box. I
also made a graph to show that the maximum cut out size is around
3.30cm.

[IMAGE]


Other than using Microsoft excel to show the maximum cut out size and
volume you can also prove it through algebra. Using algebra I can
prove the maximum volume. I already know that a 1/6 of the length or
width (which is the same, as in a square all the sides are equal)
gives me the maximum cut out size so I substituted this into the
formula.

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

(H=cut out size and x=the measurement of the side of the square)

Substitute (h=x/6)

Volume= (x-2x/6) (x-2x/6) * x/6

Multiply out



Volume= (x²-2x²/6-2x²/6+4x²/36)*x/6
===================================



Factorise
=========

Volume= x³/6-2x³/36-2x³/36+4x³/216

Volume= 36x³-12x³-12x³+4x³

[IMAGE] 216

[IMAGE]Volume= 40x³-24x³

216

[IMAGE]Volume= 16*x³

216

[IMAGE]Volume= 2*x³

27

To prove this is correct I will use this formula to work out the
maximum volume for a square with dimensions of 10cm by 10cm.

2*10³

[IMAGE] = 74.074

27

This is correct because this was the maximum volume I achieved when I
calculated it in the spreadsheets.

[IMAGE]I can also prove the maximum cut out size by using the gradient
function. The gradient function also you to the gradient at any point
on a line graph. To calculate the gradient function if y=a*x (n*a) x

For example: 3x²= (2*3) x

= 6x

This formula will give you the gradient when it equals 0. This can
find the maximum cut out size as when the gradient=0 this will give
the highest point in the graph as shown below:

I will use this function to prove the maximum cut out size for a 10cm
by 10cm square is 1.67cm:

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

= (X²-4xh+4h²) * h

= Xh²-4xh²+4h³

Substitute (h=10)

= 100x-40x+4x³

Use gradient function

0=100-80x+12x² (divide by 2) 0= 50-20x+6x²

0= 25-20x+3x²

Solve by using common formulae

I am now going to continue my investigation by looking at the shape of
rectangles. As there are too many combinations of lengths and widths
of rectangles for me to possibly even begin to investigate I am going
to investigate two different rectangles, with the ratio between the
length and width as 2:1 and 3:1.

I shall begin with a width of 20cm, and a length of 40cm, this is a
ratio of 1:2, the length being twice as long as the width.

This is the formula I put into the spreadsheet:

(2w-2x) (w-2x) x

(w= the width of the rectangle and x= the cut out size)

Below are the results I got through this spreadsheet:

Card Size: 10cm by 20cm- TO FIND EXACT CUT OUT SIZE

Width

Length

Cut out size

Volume

6.00

16.00

2.00

192.000

5.98

15.98

2.01

192.076

5.96

15.96

2.02

192.146

5.94

15.94

2.03

192.208

5.92

15.92

2.04

192.263

5.90

15.90

2.05

192.311

5.88

15.88

2.06

192.351

5.86

15.86

2.07

192.385

5.84

15.84

2.08

192.412

5.82

15.82

2.09

192.431

5.80

15.80

2.10

192.444

5.78

15.78

2.11

192.450

5.76

15.76

2.12

192.449

5.74

15.74

2.13

192.440

5.72

15.72

2.14

192.425

5.70

15.70

2.15

192.404

5.68

15.68

2.16

192.375

5.66

15.66

2.17

192.339

5.64

15.64

2.18

192.297

5.62

15.62

2.19

192.248

5.60

15.60

2.20

192.192

As you can see by the table above, the largest volume is achieved when
the cut out size of each corner of the box is 2.11cm. I also made a
graph to prove that the maximum cut out size is around 2.11cm.

OPTIMUM CUT OUT SIZE FOR BOX SIZE 10CM BY 20CM

[IMAGE]

As you can see from the graph the maximum cut out size is between
2.1cm and 2.12cm.

If I wish to work out the proportion of the box that needs to be cut
away to obtain the maximum cut out size, I need to divide 2.11cm by
10. In doing this I get an answer of 0.211, or a proportion of 1/4.73.
To see if this is correct I decided to look at different size square,
20cm by 40cm. I worked out a 1/4.73 of 20, which was 4.228cm so I made
a spreadsheet to work out the maximum cut out size and volume of a
20cm by 40cm square. I looked between 4cm and 4.3cm for the maximum
cut out size, as I knew the cut out size should be around that figure.

Card Size: 20cm by 40cm

Length

Width

Cut out size

Volume

31.60

11.60

4.20

1,538.578

31.58

11.58

4.21

1,538.607

31.56

11.56

4.22

1,538.622

31.54

11.54

4.23

1,538.624

31.52

11.52

4.24

1,538.611

31.50

11.50

4.25

1,538.585

31.48

11.48

4.26

1,538.545

31.46

11.46

4.27

1,538.491

31.44

11.44

4.28

1,538.424

31.42

11.42

4.29

1,538.343

31.4

11.4

4.3

0.000

As you can see the maximum cut out size is 4.23cm, which is around the
answer, I worked out. So this definitely proves that 1/4.73 of the
width of the box gives me the maximum cut out size of the box. I also
made a graph to show that the maximum cut out size is around 4.23cm.

Optimum cut out size For Box Size 20cm by 40cm

[IMAGE]

This graph shows that the maximum cut out size is around 4.23cm.

Before investigating the rectangles in the ratio of 3:1 I need to
prove through algebra how to get the maximum volume for the rectangle
with a ratio of 2:1. I have already worked out that 1/4.73 of the
width of the box gives me the maximum cut out size of the box, so I
have to substitute this into the volume formula.

I shall begin with a width of 10cm, and a length of 30cm, this is a
ratio of 1:3, the length being three times as long as the width.

This is the formula I put into the spreadsheet:

(3w-2x) (w-2x) x

Below are the results I got through this spreadsheet:

Card Size: 10cm by 30cm

Length

Width

Cut out size

Volume

25.60

5.60

2.20

315.392

25.58

5.58

2.21

315.447

25.56

5.56

2.22

315.492

25.54

5.54

2.23

315.526

25.52

5.52

2.24

315.550

25.50

5.50

2.25

315.563

25.48

5.48

2.26

315.565

25.46

5.46

2.27

315.556

25.44

5.44

2.28

315.537

25.42

5.42

2.29

315.508

25.40

5.40

2.30

315.468

25.38

5.38

2.31

315.418

25.36

5.36

2.32

315.357

25.34

5.34

2.33

315.285

25.32

5.32

2.34

315.204

25.30

5.30

2.35

315.112

25.28

5.28

2.36

315.009

25.26

5.26

2.37

314.896

25.24

5.24

2.38

314.773

25.22

5.22

2.39

314.640

25.20

5.20

2.40

314.496

As you can see by the table above, the largest volume is achieved when
the cut out size of each corner of the box is 2.26cm. I also made a
graph to prove that the maximum cut out size is around 2.26cm.

Optimum cut out size For Box Size 10cm by 30cm [IMAGE]

This graph shows that the maximum cut out size is around 2.26cm.

If I wish to work out the proportion of the box that needs to be cut
away to obtain the maximum cut out size, I need to divide 2.26cm by
10. In doing this I get an answer of 0.226, or a proportion of 1/4.42.
To see if this is correct I decided to look at different size square,
20cm by 60cm. I worked out a 1/4.42 of 20, which was 4.52cm so I made
a spreadsheet to work out the maximum cut out size and volume of a
20cm by 60cm square. I looked between 4.45cm and 4.55cm for the
maximum cut out size, as I knew the cut out size should be around that
figure.

Card Size: 20cm by 60cm

Length

Width

Cut out size

Volume

51.20

11.20

4.40

2,523.136

51.18

11.18

4.41

2,523.368

51.16

11.16

4.42

2,523.580

51.14

11.14

4.43

2,523.769

51.12

11.12

4.44

2,523.938

51.10

11.10

4.45

2,524.085

51.08

11.08

4.46

2,524.210

51.06

11.06

4.47

2,524.314

51.04

11.04

4.48

2,524.398

51.02

11.02

4.49

2,524.459

51.00

11.00

4.50

2,524.500

50.98

10.98

4.51

2,524.519

50.96

10.96

4.52

2,524.518

50.94

10.94

4.53

2,524.495

50.92

10.92

4.54

2,524.451

50.90

10.90

4.55

2,524.386

50.88

10.88

4.56

2,524.299

50.86

10.86

4.57

2,524.192

50.84

10.84

4.58

2,524.064

50.82

10.82

4.59

2,523.914

50.80

10.80

4.60

2,523.744

As you can see the maximum cut out size is 4.51cm, which is around the
answer, I worked out. So this definitely proves that 1/4.73 of the
width of the box gives me the maximum cut out size of the box. I also
made a graph to show that the maximum cut out size is around 4.51cm.

Optimum cut out size For Box Size 20cm by 60cm

[IMAGE]

This graph shows that the maximum cut out size is around 4.51cm.

I need to prove through algebra how to get the maximum volume for the
rectangle with a ratio of 3:1. I have already worked out that 1/4.42
of the width of the box gives me the maximum cut out size of the box,
so I have to substitute this into the volume formula.



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