# Maths Investigation on Trays.

Extracts from this document...

Introduction

Maths Investigation on Trays Introduction I am trying to find out if this hypothesis is correct. The volume of the tray is maximum when the area of the base is equal to the area of the 4 sides. I tray is made using a net made below. You cut out the squares marked x then fold together to form a tray which has a base and 4 sides. This is where the above hypothesis comes from to find this out I will have to test. Testing the Theory My plan of action is to find out if the hypothesis is correct. To do this, I will have to test it with many different sized squares and prove that the volume predicted does actual make the biggest volume. To get an accurate result, I will need to do decimal tests as well as whole numbers. Also I will do other tests on other relevant shapes like rectangles. To start, I will use this formulae to help find out the max volume of a 18 by 18 grid box x 18 x x 18 x x An 18 by 18 tray is a good starting tray because it not a square or cube number this makes itr easy to use. It also can have 1,2,3,6,9 and 18 so if this hypothesis is true it should work with this size tray. Table of Formulae x Area Base Area 4 sides Volume 1 (18-2)2 4(18-2) ...read more.

Middle

8 64 256 512 9 36 216 324 10 16 160 160 11 4 88 44 12 0 0 0 Again we see that our method is being proven and the hypothesis with the max area being when x is 6 and when the 4 sides have the same area as the base. But to make sure that this is the max volume we will do another x Area Base Area 4 sides Volume 3.2 309.76 225.28 991.23 3.4 295.84 233.92 1005.9 3.6 282.24 241.92 1016.1 3.8 268.96 249.28 1022 4 256 256 1024 4.2 243.36 262.08 1022.1 4.4 231.04 267.52 1016.6 4.6 219.04 272.32 1007.6 4.8 207.36 276.48 995.33 5 196 280 980 Here is the decimal search and this says that when x is 4 it still makes the max volume. x Area base Area 4 side volume 3.8 268.96 249.28 1022 3.9 262.44 252.72 1023.5 4 256 256 1024 4.1 249.64 259.12 1023.5 We can see here that the hypothesis is correct even within a .1 margin of the numbers This graph sows the results and we can see from these that where the base area line (coloured purple) and the line representing the area of the 4 sides (coloured yellow) cross the volume (coloured turquoise) is at its max proving the hypothesis theory right for the second time. The next size tray I will use will be 15 this one is not divisible by 6 with a whole number so it doesn't fit that pattern the last did ...read more.

Conclusion

So it should be the best option for a result that works for the hypothesis 18 x Area Base area 4 sides Volume 1 (24-2x) (18-2x) 2x(18-2x) 2x(24-2x) 1(24-2x)(18-2x) 2 (24-4x) (18-4x) 4x(18-4x) 4x(24-4x) 2(24-4x)(18-4x) 3 (24-6x) (18-6x) 6x(18-6x) 6x(24-6x) 3(24-6x)(18-6x) 4 (24-2x) (18-8x) 8x(18-8x) 8x(24-8x) 4(24-8x)(18-8x) 5 (24-2x) (18-10x) 10x(18-10x) 10x(24-10x) 5(24-10x)(18-10x) 6 (24-2x) (18-12x) 12x(18-12x) 12x(24-12x) 6(24-12x)(18-12x) 7 (24-2x) (18-14x) 14x(18-14x) 14x(24-14x) 7(24-14x)(18-14x) 8 (24-2x) (18-16x) 16x(18-16x) 16x(24-16x) 8(24-16x)(18-16x) 9 (24-2x) (18-18x) 18x(18-18x) 18x(24-18x) 9(24-18x)(18-18x) (24-2x) (18-x) 2x(18-2x) 2x(24-2x) x(24-2x)(18-2x) x Area Base area 4 sides Volume (24-2x) (18-2x) 2x(18-2x) 2x(24-2x) x(24-2x)(18-2x) 1 352 76 352 2 280 136 560 3 216 180 648 4 160 208 640 5 112 220 560 6 72 216 432 7 40 196 280 8 16 160 128 9 0 108 0 As you can see the hypothesis has not worked as I predicted. I will not do a decimal search either because the results are so far apart it will probably be waste of time. So the hypothesis does not work for rectangles, or one of these sizes. We can tell it is wrong because that the line for the 4 sides and the line which is the base value cross when the volume is not at its highest. This means the hypothesis isn't right. So in conclusion to rectangles the hypothesis does not work with rectangles because they are to different the rectangle has 2 sizes of sides while the square does not. So the hypothesis can never work because it was made for a square tray with one measurement where as the rectangle had two. ...read more.

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