 Level: AS and A Level
 Subject: Maths
 Word count: 1752
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.
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
Area = height x width
Area =) x )
=
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 wa 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
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 = x () x () sinθ
A =w2sinθ
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;
=w2sin θ = 0
Cos θ = 0
θ = 90°
SemiCircular crosssection
l = θ x r
l = θ x r
r =
∴ r =
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.
Conclusion
Question 2)
(Perks and Prestage, 2001b, p95)
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, 20032010, Debt Reduction Calculators. [online] <http://www.vertex42.com/Calculators/creditcardpayoffcalculator.html> [Accessed 03 November 2010]
The Mathematics Association, 2002. ICT and Mathematics: a guide to learning and teaching mathematics 1119, Produced for the Teacher Training Agency.
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