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Best shape for gutter and further alegbra - using Excel to solve some mathematical problems.

Extracts from this document...

Introduction

EDM 252 Using and Applying Information Technology in Mathematics 1

Assignment 1

1)  A length of guttering is to be made by taking a rectangular piece of plastic and folding up the edges.

The dotted lines symbolise the folding of the edges.image00.png

Let the measurements be units (as no specific measurements are given)  

Volume max = Area of cross section max x length

Since the length is just a multiplicative constant, maximizing the volume is the same as maximizing the cross sectional area.

Therefore the area of cross section max will determine the maximum volume of the gutter.

Using a width of 10 units, I will investigate the different types of cross sections that could be used.  The first obvious shape to construct would be a square cross section.  

Square Cross section

image01.png

image08.png

Area = height x width

Area =image15.pngimage15.png) x  image16.pngimage16.png)

          = image17.pngimage17.png

Trapezium cross section

To find the maximum volume I need to establish a relationship between the three variables θ, width of base and Volume. We can do that from the following diagram:

Area = (1/2)* h * (a + b)

where

   h  = (w - a) cos (θ)              (height)

   a =  a                                   (first parallel side)

   b = a + 2(w - a) sin (θ)       (second parallel side)

To calculate angle θ, I have taken the vertical perpendicular to the base as the starting point at θ = 0,

Firstly, due to the 3 variables I fixed a and w-a to be equal length and varied θ by 1 degree.

A snapshot of the values obtained using excel is below.

θ° with horizontal

a

b

Area

28

3.333

3.333

14.41629

29

3.333

3.333

14.42938

30

3.333

3.333

14.43376

31

3.333

3.333

14.42935

32

3.333

3.333

14.41606

Highlighted in red using conditional formatting is the greatest area using these values and varying θ

...read more.

Middle

14.08059

41

1

4.5

13.4226573

29

3.33333

3.33333

14.42938

30

3.33333

3.33333

14.43376

31

3.33333

3.33333

14.42935

The formulae to calculate the area of this cross section are given in Appendix 1.

Triangular cross section(V shape)

-

Area =image02.pngimage02.png  x (image03.pngimage03.png) x (image04.pngimage04.png) sinθ

A =image05.pngimage05.pngw2sinθ

By varying θ by 1°, where  0° is at the point of the two sides touching each other to make a straight line.  Below is a snapshot of the excel spreadsheet showing optimum results.  

θ°

w/2

w/2

Area

88

5

5

12.49239

89

5

5

12.4981

90

5

5

12.5

91

5

5

12.4981

92

5

5

12.49239

The formulae used to calculate the area of this cross section is given in appendix 2.

By using algebra I can find θ at which the optimum capacity exists.

Angle to give Optimum capacity;

image06.pngimage06.png=image07.pngimage07.pngw2sin θ = 0

Cos θ = 0

θ = 90°  

Semi-Circular cross-section

l = θ x r

l = θ x r

r = image09.pngimage09.png

 r = image10.pngimage10.png

Area of semi circle = ½ x pi x r2

θ°

radius

Area

180

3.183099

31.83099

Formulae used in excel for this calculation is given in Appendix 2.

By the use of conditional formatting of the entire spreadsheet I have found that the best cross section to use for a gutter to carry the maximum water capacity is a semi circle.

2)  

Using Excel; it will take 21 months to clear the debt.

...read more.

Conclusion

Question 2)

 (Perks and Prestage, 2001b, p95)

image14.png

a) Change the raw data in the spreadsheet to give a mean of 6.

b) Get a mean of 5 and a mode of 4.

c) Get a mean, median and mode of 5.

d) Get a mean of 4, a median of 3 and mode of 5

Question 3) (.  (Perks and Prestage, 2001a, p62)

An article in the Evening Post reads ‘The average sponsorship money collected by each child was £6.00; what an effort!’ The Evening News stated ‘An average of £5.52 – what a collection!’ Both papers showed the same diagram.  Which paper got its sums right?

In this question the pupil would use the original bar chart from the 1st question to find the averages.  There is however no indication of whether mean, median or mode is expected.  Both newspapers could have the correct information but choose to display it in a different way.  The mathematics focuses on understanding the connection between raw data and the smoothing of the data shown in representation. (Prestage and Perks, 2001a, p62)

References

Math Forum, 2003, Maximizing The Volume of a Rain gutter. [online] Available at <http://mathforum.org/library/drmath/view/64541.html> [Accessed 03 November 2010]

Perks, P., and Prestage, S., 2001a. Adapting and Extending Secondary Mathematics Activities. London: David Fulton Publishers

Perks, P., and Prestage, S., 2001b. Teaching the National Numeracy Strategy at Key Stage 3. London: David Fulton Publishers

Vertex42, 2003-2010, Debt Reduction Calculators. [online] <http://www.vertex42.com/Calculators/credit-card-payoff-calculator.html> [Accessed 03 November 2010]

The Mathematics Association, 2002. ICT and Mathematics: a guide to learning and teaching mathematics 11-19, Produced for the Teacher Training Agency.

...read more.

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