When the area of the base is the same as the area of the four sides, the volume of the tray will be a maximum.

3cm

8cm

92cm

64cm

3x8x4=96cm

4cm

6cm

44cm

36cm

4x6x4=96cm

5cm

4cm

80cm

6cm

5x4x4=80cm

6cm

2cm

24cm

4cm

6x2x4=48cm

The tray with the maximum volume is the tray with the height of 2cm. With this tray area of the base again does not equal to the area of the 4 sides.

2 x 12

Height:

Base Length:

Volume:

Base Area:

Area of 4 sides:

cm

0cm

00cm

00cm

x10x4=40cm

2cm

8cm

28cm

64cm

2x8x4=64cm

3cm

6cm

08cm

36cm

3x6x4=72cm

4cm

4cm

64cm

6cm

4x4x4=64cm

5cm

2cm

20cm

4cm

5x2x4=40cm

This table shows the results I got for the measurements of the 12cm by 12cm net. The formulas that I used were the same to what I used for the others. The maximum volume that I got for this tray was with the height of 2cm the area of the base equals to the area of the 4 sides with this tray.

7 x 7

Height:

Base Length:

Volume:

Base Area:

Area of 4 sides:

cm

5cm

25cm

25cm

x5x4=20cm

2cm

3cm

8cm

9cm

2x3x4=24cm

3cm

cm

3cm

cm

3x1x4=12cm

This table shows the results I got for the measurements of the 7cm by 7cm net. The formulas that I used were the same to what I used for the others. The maximum volume is when the height is 1cm, but the area of the base does not equal to the area of the 4 sides.

Prediction: I predict that for the 24cm by 24cm tray the maximum volume will be 4cm. From the tables which I have presented, I have noticed that the ratio is always 1/6 when the square/net and length are divided between one another. I also predict that the ratio outcome for the 24cm by 24cm tray will also be 1/6.

24 x 24

This table shows the results I got for the measurements of the 12cm by 12cm net. The formulas that I used were the same to what I used for the others.

This is also shows me that my predictions were correct as the maximum volume for the 24cm by 24cm tray was 4cm which was 1,024cm. Also the ratio outcome was 1/6.

Height:

Base length:

Volume:

Base Area:

Area of 4 sides:

cm

22cm

484cm

484cm

x22x4=880cm

2cm

20cm

800cm

400cm

2x20x4=160cm

3cm

8cm

972cm

324cm

3x18x4=216cm

4cm

6cm

,024cm

256cm

4x16x4=256cm

5cm

4cm

980cm

96cm

5x14x4=280cm

6cm

2cm

864cm

44cm

6x12x4=288cm

Height:

Base length:

Volume:

Base Area:

Area of 4 sides:

7cm

0cm

700cm

00cm

7x10x4=280cm

8cm

8cm

512cm

64cm

8x8x4=256cm

9cm

6cm

324cm

36cm

9x6x4=216cm

0cm

4cm

60cm

6cm

0x4x4=160cm

1cm

2cm

44cm

4cm

1x2x4=88cm

The maximum volume was when the height of the tray was 7cm. The area of the base though did not equal to the area of the 4 sides.

8x8=64cm

=64x2=128cm

The formula I used to calculate this was: lxbxh

Height:

Base length:

Volume:

Base Area:

Area of 4 sides:

cm

7cm

49cm

49cm

x7x4=28cm

2cm

5cm

50cm

25cm

2x5x4=40cm

3cm

3cm

27cm

9cm

3x3x4=36cm

4cm

cm

4cm

cm

4x1x4=16cm

This table shows the results I got for the measurements of the 9cm by 9cm net. The formulas that I used were the same to what I used for the others. The maximum volume is when the height is 1cm, the area of the base does not equal to the area of the 4 sides.

I then did a further investigation to find out weather the shop keepers statement was true. This time instead of cutting out square nets I cut of rectangles of different widths and lengths e.g. 12 x 6. I done two rectangles form the first rectangle I took of 1cm off each corner like I did for the squared trays and for the second rectangle trays I took off 0.5, 1, 1.5, 2 etc of each corner to find out weather doing halves I can find out if the shop keepers statement is true.

8 x 8

I started this trays investigation the same way I started the other tray investigations. I drew a net of a rectangle 18 by 8 and then cut off 0.5 etc of each of the 4 corners. I then measured the Area of the base, the area of the 4 sides and the volume. To measure the area of the base and the area of the 4 sides I used the same calculating methods that I did for the other trays for example

Lxb

16 x 6= 96cm

16 x 6 x 4=384cm

The method that I used to find out the volume was the same method used for the other trays for example

Lxbxh

16x6x1= 96cm

This is the net of the 18 by 8cm tray which I cut out.

Height:

Base length:

Volume:

Base Area:

Area of 4 sides:

6 x 6

96cm

96cm

6x6x4=348cm

2

4 x 3

84cm

42cm

4x3x4=168cm

3

4 x 4

68cm

56cm

4x4x4=224cm

4

2 x 2

96cm

24cm

2x2x4=96cm

This is a graph showing all the results for each of the headings on the top of the table and for each height. This table shows that the maximum volume is when the height is 3cm but the area of the base does not equal to the area of the 4 sides.

The after completing this tray I done another tray this time using halves to cut off the corners of the tray.

20x10

This is the net that I cut for the 15 by 5 and cut off halve e.g. 0.5, 1, 1.5 etc of each corner were it is marked on the rectangle.

Height:

Base length:

Volume:

Base Area:

Area of 4 sides:

0.5cm

6x8

64cm

28cm

6x8x0.5x4=256cm

cm

6x8

28cm

28cm

6x8x4=512cm

.5cm

5x6

35cm

90cm

5x6x1.5x4=540cm

2cm

3x4

204cm

52cm

3x4x4=208cm

2.5cm

1x2

55cm

22cm

1x2x2.5x4=220cm

3cm

9x0

27cm

9cm

9x0x4=36cm

This table shows the results that I got for cutting out each corner of the tray. The reason why I cut off 0.5 and then went on to cut off 1cm was because this why a pattern was formed for me to find out the maximum volume and are of the 4 sides. The maximum volume was when the height was 2cm, the area of the base though did not equal to the area of the 4 sides.

The method that I used to calculate the volume was

Lxbxh

e.g. 13x3x0.5=19.5cm

The formulas that are needed for these rectangle trays is only one formulas which is

X

6

Most of the trays that I made did have the area of the base equal to the area of the 4 sides with the maximum volume but not all of the trays had this theory as some of the trays did not have the area of the base equal to the area of the 4 sides with the maximum volume at all. I will be talking about this in more detail in the conclusion.

Conclusion (Observation)

For this trays investigation I investigated weather or not the shop keepers statement that 'When the volume is maximum the area of the base equals to the area of the 4 sides' to see if this statement was true or not. This investigation also showed me that my prediction that I predicted was true because for the 24 by 24cm tray the maximum volume was when the height of the tray was 3cm the prediction was also correct that I predicted because the ratio that I got for the 24 by 24 was 1/6. I done some graphs on which I plotted the results the graphs that I did were for the 18cm by 18cm, 16cm by 16cm and the rectangle which was the 20cm by 10cm. There were many trends and patterns in the graphs. The line graph for the 18cm by 18cm squared net tray showed me that what ever the maximum volume is the area of the base equals to it this is shown on the graph because the maximum volume is when 1cm has been cut off and on the graph at this point the line for the area of the base joins the line of the volume at point 1cm. There are some trends in the graph like the volume line increases its height when there is less cm cut off for example at 1cm it is at a big height but when the corners get cut of more and more the line rapidly drops i.e. when 1cm has been cut off the corner the line in red on the line graph is quite high but when the cuttings get bigger and bigger like when 7cm has been cut off the red line drops down to a lower volume. There is an odd thing also in the 18 by 18 line graph because the line for the volume and the line for the area of the base both meet up and drop form a high volume to a low volume this shows me that there are trends in the graph this also shows a kind of pattern forming but the line for the area of the 4 sides does not join up to any of the two lines and also it has got a steady line so it does not increase and then drop. This could be showing me though that I have gone wrong somewhere as it does not join up the other two lines. This graph also tells me that area of the base equals to the maximum volume. The line graph for the 16cm by 16c has got big pattern init and it is quite similar to the 18 by 18 line graph. The 16 by 16 graph is quite similar to the line graph because like on the 18 by 18 graph the lines for the volume and the area of the base also join up with the maximum volume it is also the same because when the height I small the volume is high but when the height gets higher the volume gets lower just like it did on the 18 by 18 line graph, but the area of the 4 sides on this graph is different to the one of the 18 by 18 because on this graph the line is in a semi circle shape so it cures but in the other graph the line just goes up and down form one cross to another also on this graph at a point the line for the area of the 4 sides joins the line for the volume but it does not join when the volume is maximum. By looking at this you can see that the area of the base equals to the maximum volume and that the area of the 4 sides increase only when the there is a maximum volume and then decrease. The last graph that I had was for the 20 by 10 the rectangles of which I took of halves of the corners. This graph shows again that the area of the base equals to the maximum volume because the area of the base joins up to the volume when it is at the maximum point, there is a pattern in this line graph to because the volume and area of the base increase when the height of the tray is small but then slowly the volume and area of the base start to drop when the height of the tray increase this again is similar to the other two tray graphs. The area of the 4 sides increase when the heights are small and then decrease when the heights are big, there is a odd trend because the volume and area of the base have the same pattern but the area of the base is bigger then the volume another odd thing is that the area of the 4 sides joins the top of the volume and area of the base but the rapidly drops this could have caused an anomalies in the graph and the results.

Evaluation

This investigation went really well because I got a good set of results as I followed all the guidance that I was given toughly. This investigation was also good because the prediction that I predicted was true that the maximum volume for the 24 by 24 tray will be when the height is 4cm and from the tables which I have presented, I have noticed that the ratio is always 1/6 when the square/net and length are divided between one another. I also predict that the ratio outcome for the 24cm by 24cm tray will also be 1/6, there were many other reasons why this investigation went good they were that I used the same method to calculate each volume, area of base and the area of the 4 sides therefore I got some accurate pieces of results also I got some good graphs which show all the area of the base, the maximum volume and the volumes and the area of the 4 sides they also show that the area of the base and 4 sides equal to the maximum volume. The investigation also went well because I got all the algebra formulas needed to back up my prediction and the shop keeper's statement this was another reason why this investigation went good because I found out the right information needed. If I had the was to investigate even more further to see what results I would get I would make more trays for the rectangles on which I would cut off up to 10.5 or even more half corners I also would make the trays over 24 by 24 and see if I would get the same amount of heights or bigger or smaller, I also would try to use different calculation to see weather or not I get different results. I could try to find out different formulas to use for the algebra and see weather it gave me the same results as the algebra formulas that I used now another thing that I could use is make a decimal line graph, the last thing that I would if I did further investigation is that I would ask for more help and look at different guidance sheets I also would instead of using 1cm squared paper to cut off the halves of the rectangle corners I would use half cm squared paper to get more accurate results.

11.5