Mathematics GCSE The Open Box Problem Tiers F, I and H

Introduction

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 in figure 1.

Figure 1:

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

The main aim of this activity is to determine the size of the square cut out which makes the volume of the box as large as possible for any given rectangular sheet of card.

. 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.

Question 1

I began work on question 1, which was to investigate the correct cut out size to satisfy the largest possible volume of the open box on any sized square piece of card.

I started by using a square piece of card measuring 10 x 10.

To work out the volume for the open box with the variable being 'x' I made a formula to help:

V=x(l-2x)(l-2x)

Here are my results:

size of cut out 'x' (cm)

volume (cm³)

64

2

72

3

48

4

6

The highest volume is 72cm³ with a cut out size of 2cm. I found that the highest volume must have a cut out size of between 1 and 2cm so I tried the formula for cut out sizes between 1 and 2cm:

size of cut out 'x' (cm)

volume (cm³)

.1

66.924

.2

69.312

.3

71.188

.4

72.576

.5

73.5

.6

73.984

.7

74.052

.8

73.728

.9

73.036

I again found that the highest volume is 74.052cm³ with a cut out size of 1.7cm. I then homed in again and used the formula for cut out sizes between 1.6 and 1.7cm.

size of cut out 'x' (cm)

volume (cm³)

.61

74.009124

.62

74.030112

.63

74.046988

.64

74.059776

.65

74.0685

.66

74.073184

.67

74.073852

.68

74.070528

.69

74.063236

I found that the highest volume is 74.073852 when 'x' is to two decimal places being 1.67.

I created a graph to show the results when the cut out size is to two decimal places.

Introduction

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 in figure 1.

Figure 1:

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

The main aim of this activity is to determine the size of the square cut out which makes the volume of the box as large as possible for any given rectangular sheet of card.

. 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.

Question 1

I began work on question 1, which was to investigate the correct cut out size to satisfy the largest possible volume of the open box on any sized square piece of card.

I started by using a square piece of card measuring 10 x 10.

To work out the volume for the open box with the variable being 'x' I made a formula to help:

V=x(l-2x)(l-2x)

Here are my results:

size of cut out 'x' (cm)

volume (cm³)

64

2

72

3

48

4

6

The highest volume is 72cm³ with a cut out size of 2cm. I found that the highest volume must have a cut out size of between 1 and 2cm so I tried the formula for cut out sizes between 1 and 2cm:

size of cut out 'x' (cm)

volume (cm³)

.1

66.924

.2

69.312

.3

71.188

.4

72.576

.5

73.5

.6

73.984

.7

74.052

.8

73.728

.9

73.036

I again found that the highest volume is 74.052cm³ with a cut out size of 1.7cm. I then homed in again and used the formula for cut out sizes between 1.6 and 1.7cm.

size of cut out 'x' (cm)

volume (cm³)

.61

74.009124

.62

74.030112

.63

74.046988

.64

74.059776

.65

74.0685

.66

74.073184

.67

74.073852

.68

74.070528

.69

74.063236

I found that the highest volume is 74.073852 when 'x' is to two decimal places being 1.67.

I created a graph to show the results when the cut out size is to two decimal places.