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• Level: GCSE
• Subject: Maths
• Word count: 4546

# Producing a Box

Extracts from this document...

Introduction

I was given a 12 by 12 square, and asked to reshape it into a ‘strawberry box’ type shape, to find the box with the largest volume.  Below, is a diagram of what I had to work with.

To find out the volumes of my boxes, I will cut out a 1cm by 1cm square from each corner of my 12 by 12cm square and shape it into a box.  I will then find the volume of this box by multiplying the length by the breath by the height.  I will then repeat this method again, only I will be cutting off a 2cm by 2cm square, and a 3cm by 3cm square, from each corner, until I can no longer remove any more.  I will display these shapes on the following pages, and present the figures in a table after the last shape.

Shape One

Volume = Length x Breath x Height

Length = 10cm

Breath = 10cm

Height = 1cm

Volume = 10 x 10 x 1

Volume = 100 cm³

Shape Two

Volume = Length x Breath x Height

Length = 8cm

Breath = 8cm

Height = 2cm

Volume = 8 x 8 x 2

Volume = 128 cm³

Shape Three

Volume = Length x Breath x Height

Length = 6cm

Breath = 6cm

Height = 3cm

Volume = 6 x 6 x 3

Volume = 108 cm³

Shape Four

Volume = Length x Breath x Height

Length = 4cm

Breath = 4cm

Height = 4cm

Volume = 4 x 4 x 4

Volume = 64 cm³

Shape Five

Volume = Length x Breath x Height

Length = 2cm

Breath = 2cm

Height = 5cm

Volume = 2 x 2 x 5

Volume = 20 cm³

 Shape Length Breath Height Volume (cm³) 1 10 10 1 100 2 8 8 2 128 3 6 6 3 108 4 4 4 4 64 5 2 2 5 20

By looking at this table, I can see that shape 2 has the largest volume.  To look at the possibility of there being a larger volume available, I am now going to place these figures in a graph on the next page.

After looking at the graph, I realise that there could be a larger volume available.  So I am now going to investigate this possibility.

Because shape 2 had the largest volume, and I had removed a

2cm by 2cm square from each corner to make shape 2.

Middle

Length = 10 – 3.2

Length = 6.8                                                Length = Breath

Length = 6.8

Breath = 6.8

Height = 1.6

Volume = Length x Breath x Height

Volume = 6.8 x 6.8 x 1.6

Volume = 73.984 cm³

Shape 9

Length = 10 – (Height + Height)

Length = 10 – (1.7 + 1.7)

Length = 10 – 3.4

Length = 6.6                                                Length = Breath

Length = 6.6

Breath = 6.6

Height = 1.7

Volume = Length x Breath x Height

Volume = 6.6 x 6.6 x 1.7

Volume = 74.052 cm³

Shape 10

Length = 10 – (Height + Height)

Length = 10 – (1.8 + 1.8)

Length = 10 – 3.6

Length = 6.4                                                Length = Breath

Length = 6.4

Breath = 6.4

Height = 1.8

Volume = Length x Breath x Height

Volume = 6.4 x 6.4 x 1.8

Volume = 73.728 cm³

 Shape Length Breath Height Volume (cm³) 5 6.2 6.2 1.9 73.036 7 7 7 1.5 73.5 8 6.8 6.8 1.6 73.984 9 6.6 6.6 1.7 74.052 10 6.4 6.4 1.8 73.728

By looking at my results like this, I can see that shape 9 is the largest possible box, you can make from a 10cm by 10cm cube.  Now, I want to check my previous formula against the 10 by 10cm cube.

The Formula for Finding the Volume of A Strawberry Box

x          x

x                          x

x                          x

x         x

Length = 10 – 2x

Breath = 10 – 2x

Height = x

Volume = Length x Breath x Height

Volume = (10 – 2x)(10 – 2x) x x

I will now check if my formula is correct

Let x = 3

Volume = (10 – 2x)(10 – 2x) x x

Volume = (10 – 6) x (10 – 6) x 3

Volume = 4 x 4 x 3

Volume = 48 cm³

By comparing my results here with Shape 3 on page 9, I can see that my formula has worked just fine.  However, I am going to try it out on yet another size of square (24cm by 24cm square) to be sure that it will work on sizes.

I am going to workout the figures on the next few pages, and place them in a table at the end.

Shape 1

Length = 24 – (Height + Height)

Length = 24 – (1 + 1)

Length = 24 – 2

Length = 22                                                Length = Breath

Length = 22

Breath = 22

Height = 1

Volume = Length x Breath x Height

Volume = 22 x 22 x 1

Volume = 484 cm³

Shape 2

Length = 24 – (Height + Height)

Length = 24 – (2 + 2)

Length = 24 – 4

Length = 20                                                Length = Breath

Length = 20

Breath = 20

Height = 2

Volume = Length x Breath x Height

Volume = 20 x 20 x 2

Volume = 800 cm³

Shape 3

Length = 24 – (Height + Height)

Length = 24 – (3 + 3)

Length = 24 – 6

Length = 18                                                Length = Breath

Length = 18

Breath = 18

Height = 3

Volume = Length x Breath x Height

Volume = 18 x 18 x 3

Volume = 972 cm³

Shape 4

Length = 24 – (Height + Height)

Length = 24 – (4 + 4)

Length = 24 – 8

Length = 16                                                Length = Breath

Length = 16

Breath = 16

Height = 4

Volume = Length x Breath x Height

Conclusion

Shape 1

Size of square removed = x = 1

Length    = 24 – 2x

Height     = x

Volume = (24 – 2x)(12 – 2x) X x

Vol = (24 – 2)(12 – 2) X 1

Vol = 22  X  10  X  1

Vol = 220cm³

Shape 2

Size of square removed = x = 2

Length    = 24 – 2x

Height     = x

Volume = (24 – 2x)(12 – 2x) X x

Vol = (24 – 4)(12 – 4) X 2

Vol = 20  X  8  X  2

Vol = 320cm³

Shape 3

Size of square removed = x = 3

Length    = 24 – 2x

Height     = x

Volume = (24 – 2x)(12 – 2x) X x

Vol = (24 – 6)(12 – 6) X 3

Vol = 18  X  6  X  3

Vol = 324cm³

Shape 4

Size of square removed = x = 4

Length    = 24 – 2x

Height     = x

Volume = (24 – 2x)(12 – 2x) X x

Vol = (24 – 8)(12 – 8) X 4

Vol = 16  X  4  X  4

Vol = 256cm³

Shape 5

Size of square removed = x = 5

Length    = 24 – 2x

Height     = x

Volume = (24 – 2x)(12 – 2x) X x

Vol = (24 – 10)(12 – 10) X 5

Vol = 14  X  2  X  5

Vol = 140cm³

As before, I will place the results for finding the volume of a rectangular box with a net of size 12 by 24cm, in a table displayed below.

 Shape Length Breadth Height Volume (cm³) 1 22 10 1 220 2 20 8 2 320 3 18 6 3 324 4 16 4 4 256 5 14 2 5 140

To make this information more presentable, I am going to put it into a graph below.

I have learned from working out the largest volume using the square, that the first table/graph doesn’t always show that largest volume of box.  This table and graph shows that the largest box has been found by removing a 2cm square from each corner of the original rectangle.  I am now going to check weather or not Shape 2 is the largest shape.  Checking this will be done by removing a 1.9cm square from each corner of the original rectangle to make Shape 6, and removing a 2.1cm square from each corner of the original rectangle to make Shape 7.

Shape 6

Size of square removed = x = 1.9

Length    = 24 – 2x

Height     = x

Volume = (24 – 2x)(12 – 2x) X x

Vol = (24 – 3.8)(12 – 3.8) X 1.9

Vol = 20.2  X  8.2  X  1.9

Vol = 314.716cm³

Shape 7

Size of square removed = x = 2.1

Length    = 24 – 2x

Height     = x

Volume = (24 – 2x)(12 – 2x) X x

Vol = (24 – 4.2)(12 – 4.2) X 2.1

Vol = 19.8  X  7.8  X  2.1

Vol = 324.324cm³

I am now going to place these two new figures in a table along with Shape 2 so that I can compare my new results with my last ones.

 Shape Length Breath Height Volume (cm³) 2 20 8 2 520 6 20.2 8.2 1.9 314.716 7 19.8 7.8 2.1 324.324

Barry                 McNickle

12 E

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