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  • Level: GCSE
  • Subject: Maths
  • Word count: 2729

volumes of open ended prisms

Extracts from this document...

Introduction

Part 1

For part 1 of this piece of math coursework I will be investigating volumes of prisms, which can be made from a 24cm, by 32cm piece of card. I will be trying to determine which shape will make the prism with the largest volume. To do this, I will be exploring the volumes of triangular prism, cylinders, quadrilateral prisms, pentagonal prism, hexagonal prism, heptagonal prism and octagonal prism. I will then try to work out a formula for working out the volume of an “n” sided shape.  

image00.png

image01.png

image30.png

Triangular prisms

        First, I will be investigating the volume of triangular prisms. We know that in a triangle, the lengths of the left and right sides must add up to more than the length of the base and we also know that the volume of any prism is the area of cross section multiplied by the length.

image37.pngimage58.pngimage40.pngimage20.pngimage36.png

                                               To find the volume, we must first find the area of the image63.pngimage70.png

7cm                                         cross-section. To find the area of a triangle we must                                                     image02.png

          h                                       use the formula Area(a) = base (b) x height (h)image08.png

2image16.pngimage20.pngimage25.png

   10cm                32cm                    To work out the height we must use                                              

                                                        Pythagoras’s theorem

image26.png

                           Using the rules of Pythagoras, we know that the height2 = 72-52

         H     7cm    therefore, the height is 4.899cm2

                           So to find the area we do: 4.899 x 10 (base) image27.png

            5cm                                                             2                    = 24.49cm 3

...read more.

Middle

image61.pngimage34.pngimage61.png

                                          Area of cross section = 3 x 9 = 27cm2

                                             Volume of prism = 27 x 32 = 864cm3image62.png

  3cm image64.png

                                32cm

            9cm

                                               Area of cross section = 2 x 10 = 20cm2image66.pngimage34.pngimage65.pngimage29.png

        32cm                                     Volume of prism = 20 x 32 = 640cm3

image67.pngimage68.png

2cm        

              10cm        

I have also worked out a formula to work out the volumes of quadrilateral:

image56.pngimage56.pngimage13.pngimage25.pngimage69.png

        c                           V=                        length of prism. Multiplies         image71.png

                                                          with area to gives us the volume. image72.pngimage73.png

image74.png

       a

                                This gives us the area of the cross section

            b

To show that this formula will give me the volume of a quadrilateral, I must put the figures for one of my quadrilaterals in and check to see whether or not the formula gives me the correct volume.

A = 6cm, B= 6cm, C = 32cm    

6 x 6 x 32 = 1152cm3

1152cm3 isthe same volume as I previously worked out, and therefore proves that this formula is correct.

I have realised that the square is the quadrilateral give the largest volume. Taking into account both the results I have obtained from the triangular and quadrilateral prisms, I conclude that regular shapes give larger volumes than irregular shapes do, therefore I will now only work out the volumes for various regular shapes.  

        To prove that infact the square will give us the largest volume; I will consider values close to the dimensions of the square.

Value no.1

Value no.2

Value no.3

Value no.4

A

5.9

5.8

5.7

5.6

B

6.1

6.2

6.3

6.4

Volume

1151.68cm3

1150.72cm3

1149.12cm3

1146.883

   32cmimage13.pngimage25.pngimage04.pngimage04.png

A

        B

...read more.

Conclusion

         Because we know that regular shapes give us the largest volume I will only work out the volumes for regular prisms.

Squares

I will work out the volumes for the various squared, by using the formula I worked out in part 1:

AB x C

     c        

            a

                b

Length of prism (cm)

Perimeter (cm)

Length of sides (cm)

Volume (cm3).

1

1200

300

90000

2

600

150

45000

3

400

100

30000

4

300

75

22500

5

240

60

18000

6

200

50

15000

Equilateral triangle.

        I will also use the formula I previously formed to work out the volume of the triangular:

             y

 a                c                                

b  

   c2  -     b2    x   b  x y  

             2 _                

         2        

Length of prism (cm)

Perimeter of triangle (cm)

Length of sides (cm)

Volume (cm3).

1

1200

400

69282.0323

2

600

200

43641.01615

3

400

133.3

26628.32952

4

300

100

17320.50808

5

240

80

13856.40646

6

200

66.6

11523.92292

Icosagon

        Out of the polygons I investigated, the 20 sided regular icosagon had the largest volume, therefore I will now work out the volume for an icosagon using the formula for an ‘n’ sided shape:

  P/2N  

Tan180     x   PL  

        N                                

           2

Length of prism (cm)

Perimeter of icosagon  (cm)

Volume (cm3).

1

1200

113647.5273

2

600

56823.76363

3

400

37882.50909

4

300

28411.88182

5

240

22729.50545

6

200

18941.25454

From this whole investigation, I have found out that a cylinder will give us the largest volume, this may be due to the fact that it is has a smooth curve rather than many sides. Now, I understand that as the Perimeter of a prism increases and the length decreases the volume increases. So, if I were to make a container that would be able to carry the largest amount of water, I would take into account the fact that cylinders with a large perimeter yet small length will give the largest volume.    

...read more.

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