Maths Investigation on Trays.

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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

8

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

(18-2)2

4(18-2)

(18-2)2

2

(18-4)2

8(18-4)

2(18-4)2

3

(18-6)2

2(18-6)

3(18-6)2

4

(18-8)2

6(18-8)

4(18-8)2

5

(18-10)2

20(18-10)

5(18-10)2

6

(18-12)2

24(18-12)

6(18-12)2

7

(18-14)2

28(18-14)

7(18-14)2

8

(18-16)2

32(18-16)

8(18-16)2

9

(18-18)2

36(18-18)

9(18-18)2

Here is the algebra I am using to find out the different results for the three table headings. The algelbra here is easy to figure out, it has no tricky parts

Table of results for Grid 18 by 18

x

Area Base

Area 4 sides

Volume

256

64

256

2

96

12

392

3

44

44

432

4

00

60

400

5

64

60

320

6

36

44

216

7

6

12

12

8

4

64

32

9

0

0

0

Here we see that when x is 3 you get the best volume. But to do a better more accurate reading into the biggest volume I will need to go in to the decimals between 2 and 4 we use a broad range because it maximum volume good lie between 2 and 3 or 3 and 4. Also you should notice that if the maximum volume is still to be 3 that would mean that the hypothesis is correct for this tray because the base area equals the area of the other 4 sides they are both 144.
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the graph shows the results for the 18 by 18 tray. As is shown where the yellow line (area of 4 sides) crosses the purple line (area base) we have the highest volume of the turquoise line.

x

Area Base

Area 4 sides

Volume

2.2

184.96

119.68

406.91

2.4

174.24

126.72

418.18

2.6

163.84

133.12

425.98

2.8

153.76

138.88

430.53

3

144

44

432

3.2

134.56

148.48

...

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