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  • Level: GCSE
  • Subject: Maths
  • Word count: 1790

A Builder has to make drains from a sheet of plastic measuring 2m x 50cms. He finds the semi-circle produces the best drain. Prove This.

Extracts from this document...

Introduction

PROBLEM:

A Builder has to make drains from a sheet of plastic measuring 2m x 50cms. He finds the semi-circle produces the best drain. Prove This.

INTRODUCTION:

I will hope to prove in this coursework that a semi-circle will make the best drain for a builder. I will show this by calculating the area and volume of the semi-circle and other shapes. Hopefully I will find that the semi-circle will give the best area which will mean that it gives a bigger volume and will therefore hold more.

image00.png

image08.pngimage22.pngimage08.pngimage01.pngimage01.png

1.

2.

PROBLEM: Which way would be the best to bend the plastic?

SOLUTION: I would probably choose diagram 2 because it is the longest shape when bent and we would not need as many drains. The reason for me not choosing diagram 1 is because I would need to many pieces of drain to go around the whole house and it would hang to low and look unsightly.

WHICH SHAPES WILL I EXPLORE?

  1. Semi-Circle

This Shape is in 2D as if you were looking at it from straight on.

This is how it looks in 3D and how it will look on a building.

  1. Rectangle

This is the shape in 2D as if you were looking at it straight on.

...read more.

Middle

25cm

54.265cm

175

25cm

27.136cm

*BOLD indicates turning point.

Trapezium

By looking at the trapezium I can see that it is made up of two triangles and a rectangle.image11.pngimage13.pngimage10.pngimage09.pngimage11.pngimage11.pngimage11.pngimage07.pngimage06.pngimage11.pngimage12.png

We found B as 25cm because when we looked at the findings from the rectangle 25cm gave the best area for our shape.

To find the area for the trapezium I am not going to change the lengths of the side instead I will change the angle θ. The values for the angles can range from 1 to 89. For calculating each of my angle sizes I will go up in steps of 5.

To find the area of the trapezium I will use the formula:

Area of trapezium = ½ the sum of the parallel sides by the      

                                  Perpendicular height.

image14.png

image15.png

image17.pngimage24.pngimage23.pngimage21.pngimage16.pngimage21.pngimage25.pngimage20.pngimage16.pngimage21.pngimage07.pngimage21.pngimage19.pngimage21.pngimage18.pngimage17.png

Working Out For Trapezium

To find X we use the equation:

Sin45 = X/12.5

X = 12.5 Sin45image26.png

X = 8.8388cms

To find Y we use this equation:

Y = 2X+25

    = 2(8.8388)+25

    = 42.6776

To find H we use the equation:

Cos45 = H/12.5

H = 12.5 Cos45

H = 8.8388cms

...read more.

Conclusion

Area of test-tube = Area of Semi-circle + area of rectangle

= 366.897 + 15.287

= 382.184

Height of Rectangle (side) cm

Circum. Of semicircle

Cm

Radius of the semicircle

Cm

Area of semicircle

Cm2

Diameter

Cm

Area

Cm2

Total Area

     Cm2

1

15.29

7.645

367.04

30.58

30.58

397.62

2

14.65

7.325

336.96

58.6

117.2

454.16

4

13.38

6.69

281.07

107.04

428.16

709.23

6

12.10

6.05

229.86

145.2

871.2

1101.06

8

10.83

5.415

184.14

173.28

1386.24

1570.38

10

9.55

4.775

143.19

191

1910

2053.19

12

8.28

4.14

107.64

198.72

2384.64

2492.28

14

7.01

3.505

77.15

196.28

2747.92

2825.07

16

5.73

2.865

51.55

183.36

2933.76

2985.31

18

4.46

2.23

31.23

160.56

2890.08

2921.31

20

3.18

1.59

15.88

127.2

2544

2559.88

22

1.91

0.955

5.73

84.04

1848.88

1854.61

24

0.64

0.32

0.64

30.72

737.28

737.92

Conclusion

Now that I have investigated all five shapes I can now display my findings in order in the table below. The shapes are listed from the one with the largest area/volume to the one with the smallest. The one with the largest area/volume is the best shape for a drain.

Stage

Area

1

392.5

5

382.184

2

312

4

299.09

3

289.389

I can now see that the best drain is the semi-circle as it has the greatest area and volume.

Extension

If I had more time I would have been able to investigate after shapes and went on to investigate about different sizes of plastic and how they effect the area and volume of the drain.

Evaluation

I found this project very rewarding and I thoroughly enjoyed it. I was pleased when at the end I found that the semi-circle was the shape that gave the greatest area and volume.

             Although I enjoyed most of the project I found some parts quite difficult, for example the area of the trapezium had a lot of complex working out to do.

             There are parts of the project that I could have made more accurate by rounding to a higher decimal place or using the Π button on the calculator.

...read more.

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