The Open Box Problem

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Part 1: Square Ryan Simmons

For any sized square sheet of card investigate the size of the cut out square that makes an open box of the largest volume.

The first square to be tested has measurements of 30 x 30 cm. For cut sizes I will start from the smallest (whole) number possible (1cm) I will then work my way up to find which size cut gives the box the largest volume.

So far as the size of the cuts increase the volume increases. I predict that a cut size of 3 x 3 cm will give an even bigger volume for the box.

A cut of 3 x 3 cm gives a volume of 1728 cm3, hence my prediction was right.

3cm x 24cm x 24cm = 1728 cm³

The prediction was right and so far there is no obvious pattern between the cut sizes and the volume of the box. To save me from drawing a diagram for every cut size I will record my results in a table

Cut size (cm)

W

L

V (cm3)

4

22

22

936

5

20

20

2000

6

8

8

944

7

6

6

792

8

4

4

568

This is a spreadsheet, where the value of the volume is a product of the cut size, the width and the length. The formula used in the spreadsheet is:

V = Cut size x W x L (on the spreadsheet the formula I used was: F6 = C6 * D6 * E6. The volume (V) being F6, the length being E6, the width being D6 and the height being C6).

The volume of the box increased with increasing cut size until the cut became 5cm, it then decreased. There is no point investigating cuts any bigger than 5 cm because after 5 cm the volume starts to decrease. From the table, 5 cm makes the largest volume so far. It may not be the biggest as I haven't tried decimals yet. Noting that the volume decreased once the cuts reached 5cm I will start to calculate the volume of cuts at 4.4cm as the volume could start decreasing before OR after 5cm.
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Cut size (cm)

W

L

V (cm3)

4.4

21.2

21.2

977.536

4.5

21

21

984.5

4.6

20.8

20.8

990.144

4.7

20.6

20.6

994.492

4.9

20.2

20.2

999.396

5

20

20

2000

5.1

9.8

9.8

999.404

5.2

9.6

9.6

997.632

I used the same formula for this spreadsheet as the other. The decimal place theory wasn't right. 5cm is still the cut, which ...

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