The Open Box Problem

Authors Avatar

The Open Box

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.

                                                  10

 

 

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 which makes the volume of the box as large as possible for any given rectangular sheet of card.

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

  1. For any sized rectangular sheet of card investigate size of the cut out square corner which makes an open box of the largest volume.

We will use a ten-cm square. We are using a square because it is easier to use one variable because a squares width and length are the same.

Therefore we can see that the maximum box area is made from the cut size of 2cms. Now I will try between 1-2 cm’s.

The highest volume is 1.7, now i will try the maximum volumes between 1.6 and 1.7.

The maximum volume for 10 by 10 cm’s squared is 1.67. Now i will try between 1.66 and 1.67.

Therefore, we can tell that the maximum volume is 1.666.

Now i will draw a graph to show the change in volume.

 

The maximum volume of this box is 1/6 of the length.

We can tell this because 10 divided by 6 is 1.6666 reoccurring.

 Now I will try a 15 by 15 square. I predict that the maximum volume will be a cut size of 2.5

 I will try between 2 and 3 because this is where the volume is at the maximum.

Now I will try between 2.41 and 2.6

As we can tell 2.5 is the maximum cut size for a 15 by 15 square. This shows my prediction was correct. 15 divided by 6= 2.5.

To help find the maximum volume of a square I should try and find the proportion of the box that needs to be cut out to find the maximum volume, I will need to divide the cut out size by the original length of the box. So

Join now!

                                 

L= Original Length

Y= Original Length- 2x

10= 2x+y or y=10-2x

X=10/6 or L/6

To make an equation I am going to substitute these into a simple equation.

V = (L – 2L/6)(L – 2L/6) L/6

Then I multiplied the 2 in each bracket with L/6 which gives-

V = (L – 2L/6) (L – 2L/6) L/6

Then I will multiply ...

This is a preview of the whole essay