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

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 w-a to be equal length and varied θ by 1 degree.

A snapshot of the values obtained using excel is below.

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

Then I varied base a which also varied w-a.  Base a was varied by 1 unit each time.

Below is a snapshot of the excel spreadsheet showing the optimum values obtained with varying a and θ.  Highlighted in red is the maximum area which would give the maximum volume for this cross section.

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 =w2 sinθ

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.

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.

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