Investigate the size of the cut out square, from any square sheet of card, which makes an open box of the largest volume.

In the graph on the next page, you can see that it is curved. This is because that the volumes increase until it reaches the highest point, and then it starts to decline after 5cm by 5cm. I could also use my graphs for comparison as well.

This is my second investigation:

8cm by 18 cm piece of card

In the table above, you can that the square cut-out that produces the largest volume is 3cm by 3cm. This produces a volume of 864cm cubed.

As I did in the last investigation, I checked to see if the decimals around 3cm (2.9 and 3.1) could produce a larger volume, but they didn't.

On the second graph (on the next page) that I've drawn, there is the same shaped curve as there was in the first, so there must definitely be a connection between the two investigations.

Now, I shall do some calculations to check for any other connections.

864 cm cubed divided by 18cm = 48 - No connection

8cm divided by 864cm cubed = 0.02083333333 - No connection

8cm divided by 3cm = 6 - That's same number I got in the last investigation, when I divided the length of the piece of card by the size of the square cut-out.

My calculations tell me that the cut-out could be a sixth of the length of the card. (X)

So my formula for working out the cut-out is

Now, I'm going to prove my theory by doing a calculation.

30cm divided by 6 = C

5 = C

My theory is correct, because in my first investigation, the 30cm by 30cm piece of card, the highest volume was produced by 5cm x 5cm square cut-out.

Task 2

I'm going to investigate the size of the cut-out square, from any sized rectangular piece of card that makes an open box of the largest volume.

As I did in the previous task, I will do two investigations and if I come up with a theory, I will prove it by doing another set of calculations. As I also did in the previous task, I will try to find a connection between the size of the square cut-out and the size of the rectangular piece of card it is being cut from.

This is a diagram of the rectangular piece of card:

To work out the volume of this piece of card here is the following formula to do so.

Volume = length x width x height

V= (X-2C) x (2X-2C) x C

V= (X-2C) x (2X-2C) C

This is my first investigation:

32cm by 16cm piece of card

If you look at the table above, you will see that the largest volume, 780cm cubed, was produced by the 3cm by 3cm square cut-out.

As I have done in the two previous investigations, I have checked to see if it is possible that the volume could get any higher, but once again it didn't.

I'm going to do a few calculations to see if there is any connection between the square cut-out and the size of the piece of card.

32 divided by 3= 10.66666 (recurring)

6 divided by 3= 5.333333 (recurring)

3 divided by 32= 0.09375

3 divided by 16= 0.1875

This is my second investigation:

20cm by 10cm piece of card

In the table above, you can see the largest volume produced (1.92.4499cm cubed) was by the 2.111 (recurring) sized square cut-out.

As you may see in the table above, I checked around what I thought was the largest volume, and found out it wasn't the highest. So I kept on checking the decimals, until I realised that the volume would keep on continuously increasing as more 1's were added on the end of 2.11.

I'm going to do some more calculations.

20 divided by 2.111(recurring)= 9.4736842105263157894736842105268

0 divided by 2.111 (recurring)= 4.7368421052631578947368421052634

2.111 (recurring) divided by 20= 0.1055555555555555555555

2.111 (recurring) divided by 10= 0.21111111111111111111111

As you evidently see, there is no connection between this one and the last.

This is my third investigation:

40cm by 20cm piece of card

In the table above, you can see the largest volume produced (1539.6cm cubed 2.d.p.) was by the 4.2222c, (recurring) sized square cut-out.

As you may see in the table above, I checked around what I thought was the largest volume, and found out it wasn't the highest. So I kept on checking the decimals, until I realised that the volume would keep on continuously increasing as more 2's were added on the end of 4.222.

Now I'm going to do my last set of calculations.

40cm divided by 4.22 (recurring)= 16.140351

20cm divided by 4.22 (recurring)= 4.7368421

4.22 (recurring) divided by 40cm= 0.1055555

4.22 (recurring) divided by 20cm= 0.2111111

When I first started this task, I thought there must be some kind of connection or relationship between the maximum volume square cut-out and the size of the piece of the card, as on the rectangle was only double the square. But I can't find it.

But what I did notice was that on investigations 2 + 3, when you divided the size of the square cut-out by the width of the piece of card and you got the same number which was 0.1055555. So there must be some kind of connection but I am unable to find.