Maximum Box Investigation

Now that we have the area, it is easy to evaluate the maximum volume in a variety of different shapes. First of all, I'm going to try a perfect cube, which i predict to be the best shape. With no lid, the cube has 5 sides, so the 576cm² must be divided by 5. This gives us 115.2cm² for each of the cube's faces. The square root of 115.2cm² is 10.73cm, and this gives us the length of each of the edges in the cube. To find the volume of this particular cube, we need to cube 10.73 which gives us 1235.37. The volume of this cube is 715.575 cm³. However, the question instructs to fold and cut, which will need a list of results in a table:

Table for card 24cm x 24cm

Length of the side of the corner square (cm)

Dimensions of the open box (cm)

Volume of the box (cm³)

22 x 22 x 1

484

2

20 x 20 x 2

800

3

8 x 18 x 3

972

4

6 x 16 x 4

024

5

4 x 14 x 5

980

6

2 x 12 x 6

864

7

0 x 10 x 7

700

8

8 x 8 x 8

512

Table for card 20cm x 20cm

Length of the side of the corner square (cm)

Dimensions of the open box (cm)

Volume of the box (cm³)

8 x 18 x 1

324

2

6 x 16 x 2

512

3

4 x 14 x 3

588

4

2 x 12 x 4

576

5

0 x 10 x 5

500

6

8 x 8 x 6

384

Table for card 15cm x 15cm

Length of the side of the corner square (cm)

Dimensions of the open box (cm)

Volume of the box (cm³)

3 x 13 x 1

69

2

1 x 11 x 2

242

3

9 x 9 x 3

243

4

7 x 7 x 4

96

5

5 x 5 x 5

25

Investigate the situation when the card is not square. Take rectangular

cards where the length is twice the width (20 x 10, 12 x 6, 18 x 9 etc.) Again, for the maximum volume is there a connection between the size of the corners cut out and the size of