Part 2 Below there is a table which shows the results from the first part of my investigation in Part 2. Firstly I investigated the maximum volume of a box made from a 15 x 15 cm piece of paper. You will see that the maximum volume of a box made from a 15x15cm piece of paper is 243 cm³. But again I wanted to find if the volume could be greater if the corner square length also had a decimal place. The table to show my results is at the bottom of this page.
At first I tried out the corner square length being in between 3 and 4 cm but this did not give me a larger volume (as you can see from the table on the bottom of the previous page. Once I got down to the length of the corner square being 3.05 cm I decided that I would not be able to find a volume larger than the 243 cm³ if the corner square length was in between 3 and 4cm. I came to this conclusion as if the volume was going to increase then the corner square length would have to be either 3.04, 3.03, 3.02 or 3.01cm and the volume would have to increase by least 2 cm³, which is impossible when the only numbers left to try were between 3 and 3.05. So, instead I decided to try numbers between 2 and 3. This proved to be more successful and I found that if the corner square length was 2.5 then the volume of the box would be 250 cm³. In the table below I have showed the results I got when I investigated on the corner square length being between 2 and 3 cm.
As you can see I started by trying the corner length being 2.5 cm and I found that this would give me a volume of 250 cm³ (higher than the volume I had got before.) Next, I tried out 2.75 cm being the length of the corner square and found that it gave me a lower volume. The volume was also lower when I tried the corner square length as 2.25 cm, 2.3 cm, 2.4 cm, 2.45 cm, 2.6 cm and 2.7cm. So, therefore, I discovered that the corner square length should be 2.5 cm to get the maximum volume of a box made from a 15 x 15 cm piece of paper.
On the table below you will see the results for my investigation on the maximum volume of a box made from a 24 x 24 cm piece of paper. If the corner square length was a whole number the maximum volume I got was 1024 cm³. However I wanted to find out if the volume would rise if I also included decimals in my corner square length. My results are in the table at the bottom of the page.
As you can see from the table on the bottom of the previous page I didn’t find a greater volume for the box. I tried a lot of different corner square lengths and I didn’t find anything. So, the maximum volume for a box made from a 24 x24 cm piece of paper is 1024 cm³.
The next size of paper I used was 10 x 10 cm. In the table below you will see the results I got. The first table shows the all the different volumes I got when the length of the side of the corner square was a whole number. The second shows the different volumes I got when the length of the side of the corner square also had decimal places.
As you can see the maximum volume for a box made from a 10 x 10 cm piece of paper is 72cm³ (if the corner squares length is a whole number.) Once again I wanted to try out whether the volume would increase if the corner square length had decimal places. The results are in the table below.
From the table on the last page you can see I found a way of getting the box to have a higher volume. First I tried the corner square length being between 2 and 3. I found that I couldn’t find a higher volume there and so I tried the corner squares length being in between 1 and 2. This was more successful and the first length I tried (1.5cm) already gave me a higher volume. The volume of the box when the corner square length was 1.5 cm was 73.5 cm³. I tried a few other corner square lengths that were between 1 and 2. The first was 1.75 cm and for this length I got a higher volume again! I got a volume of 73.9 cm³. I also tried the corner square length being 1.8, 1.25, 1.3 and 1.4 but none of these numbers gave a higher volume when they were the corner square length. Therefore, the maximum volume of a box made from a 10 x 10 cm piece of paper is 73.9 cm³.
The last size of paper I tested was a 36 x 36 cm piece of paper. First of all (like with all the other boxes) I have shown all the different volumes I got when the corner square length is a whole number. The results are in the first table below.
From the table you will see that I found that the highest volume for this box was 3456 cm³ (if the corner square length is a whole number.) I also wanted to see if I could get a higher volume if the corner square length also included decimal places. My results for this are in the table below.
From the table above you will see that I didn’t manage to find a higher volume for the box even when the corner square length included decimal places. I tried the corner squares length being in between 6 and 7 and 5 and 6 but still I didn’t find a way of increasing the boxes volume. So, the maximum volume of a box made from a 36 x 36 cm piece of paper is 3456 cm³.
Explanation of results for part 1 and 2
Part 3 I tried a lot of different methods to try and find out if the there was a connection between the original piece of paper and the size of the corners cut out. Eventually I found that if you divided the length of the side of the original piece of paper (i.e. 20cm, 15cm, 24cm, 36cm and 10cm) by the corner square length that gave you the highest volume you would get around 6. To a few of the calculations the answer was just a bit above or below 6 but if you round the answer to the nearest whole number I got 6 for each size of paper. This must mean that the length of the side of the original piece of paper is around 6 times larger than the corner square length. The table below shows my data.
Using a similar method to the previous one I found another connection. I found that the area of the original piece of paper is around 8 to 9 times bigger than the area of the 4 corners that are cut out. This means that when you divide the area of the original piece of paper by the area of all four of the corners cut out you get around 8 or 9. The table below shows my data.
Conclusions
To conclude, I found that the maximum volume for a box made from a 20 x 20 cm sheet of paper is 592.6 cm³. I also found the maximum volume for a box made from a 15 x 15 cm piece of paper is 250 cm³. Another thing I found was that the maximum volume for a box made from a 24 x 24 cm piece of paper is 1024 cm³. Some other things I found were that the maximum volume for a box made from a 10 x 10 cm piece of paper is 73.9 cm³ and the maximum volume of a box made from a 36 x 36 cm sheet of paper is 3456 cm³. I discovered that there is a connection between the original sheet of paper and the corners that are cut out. There are two connections; the first is that the length of one side of the original piece of paper is 6 times larger than the length of one side of the corner square. The second is that the area of the original sheet is 8 to 9 times larger than the area of the 4 corners that are cut out.
Reflection and Evaluation
The aim of this investigation was to find the largest possible volume of 20x20 cm, 15x15 cm, 24x24 cm, 10x10 cm and 36x36 cm pieces of paper. I also wanted to see if there was a connection between the original piece of paper and the corners that are cut out. I managed to do all these things and so fulfilled my aim. I generated my results by first of all working out the volume for each possible corner square length (that was a whole number) for each box. Once I had done this I started trying to find if there was a higher volume if the corner square length included decimals. I didn’t just try any corner square lengths with decimal places; I tried out numbers that were either a little higher or a little lower than the corner square length that had given me the highest volume before. Therefore, I think I was systematic as I did this for each box. In this project I used many skills. I used my calculator skills, my division skills, my multiplication skills, my skills at finding patterns, my problem solving skills, my trial and improvement skills, my skills at finding volume, my skills of finding area and my table drawing skills. I used my calculator skills when I was finding the volume of the different boxes and when I was trying to find patterns. I used my division skills in part 3 as I found patterns by (for example) dividing the length of one side of the original piece of paper by the length of the side of the corner square. I used my multiplication skills when I was finding the volume of boxes and finding the area of the original piece of paper (in part 3.) I used my pattern finding skills and problem solving skills in part 3 when I was trying to find a connection between the original piece of paper and the corners that were cut out. I used trial and improvement skills in part 1an 2 when I was trying to find the maximum volume of the different boxes. I used my skills at finding volume in part 1 and 2 when I was finding the maximum volume of the different boxes. I used my skills at finding area in part 3 when I found that the area of the original piece of paper is 9 times as big as the area of all 4 corners that were cut out, and I used my table drawing skills through the entire project as we had to show all of our results in tables. I don’t think I learned any new skills but I do think I did improve my old ones. I don’t really think I learnt any new concepts either. I did find some general rules. I found that the length of one side of the original piece of paper is around 6 times larger than the length of one side of the corner square. I also found that the area of the original piece of paper is about 8 to 9 times larger than the area of the 4 corner squares.
