# 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.

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Introduction

Maths Coursework Algebra-Investigating Trays Statement: The shopkeeper says, "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." Aim: To prove the shopkeeper's statement true. Task: To investigate this claim and investigate further. 18 x 18 I firstly started my trays investigation by drawing a net for a square measured 18cm by 18cm. I then cut out this net square, after it had been cut out I cut off 1cm off each of the four corners. This was so the tray would join together. I then calculated the area of the base. The formula I used to calculate the area of the base was: lxb For example: Area of Base, tray 1 16x16=256cm After this I then went on further to find the area of the four sides, as I had used this square before. I then used the formula lx4 to find the areas of the four sides, for example: 16x4=64cm I then calculated the volume of the tray. The calculations I used to find out the volume of the tray were: 14x14=256cm = 256x2=512cm The formula I used to calculate this was: lxbxh. Height: Base Length: Volume: Base Area: Area of 4 sides: 1cm� 16cm� 256cm 256cm 1x16x4=64cm 2cm� 14cm� 392cm 196cm 2x14x4=112cm 3cm� 12cm� 432cm 144cm 3x12x4=144cm 4cm� 10cm� 400cm 100cm 4x10x4=160cm 5cm� 8cm� 320cm 64cm 5x8x4=160cm 6cm� 6cm 216cm 36cm 6x6x4=144cm 7cm� 4cm 112cm 16cm 7x4x4=112cm 8cm� 2cm 32cm 4cm 8x2x4=64cm The tray that has the maximum volume is tray 3 and the area of this base equals to the area of the 4 sides which shows that the shop keeper's statement is true. ...read more.

Middle

Height: Base length: Volume: Base Area: Area of 4 sides: 1 16 x 6 96cm 96cm 16x6x4=348cm 2 14 x 3 84cm 42cm 14x3x4=168cm 3 14 x 4 168cm 56cm 14x4x4=224cm 4 12 x 2 96cm 24cm 12x2x4=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 16x8 64cm 128cm 16x8x0.5x4=256cm 1cm 16x8 128cm 128cm 16x8x4=512cm 1.5cm 15x6 135cm 90cm 15x6x1.5x4=540cm 2cm 13x4 204cm 52cm 13x4x4=208cm 2.5cm 11x2 55cm 22cm 11x2x2.5x4=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 ...read more.

Conclusion

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 ...read more.

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