Maths Coursework

Investigation

        The coursework problem set to us is to find the shape of a gutter that generates the maximum flow of water and that can contain the most water in it. The plastic used for the gutter can be no more than 30cm in width.

Plan

       To find the gutter that produces the biggest flow of water I must investigate different shapes of gutters to find which generates the largest volume. To do this I will use a variety of different shapes. I will firstly investigate a rectangle shaped gutter changing the length of the sides to form the largest volume. This is the simplest shape with a small amount of sides. The second I will investigate will be an isosceles triangle shaped gutter. The third shape I will investigate will be a trapezium shaped gutter and lastly I will investigate a semicircular shaped gutter. In calculating the trapezium and the triangle I will be using angles and length. This will take many trigonometry calculations. So rather than using a pen and paper methods to calculate these I will use spreadsheets typing in formulas to calculates answers which will only involve myself typing in the values of the shape.

Spread sheets for shapes that change in angle

Side : In this box the length of the side of the shape is put.

Angle : Into this box the angle that the side forms with the perpendicular height is put.

                         15

So the triangle and the rectangle doubled formed the same shape but were divided in different ways.

Trapezium Shaped gutters

My plan is to change the angle, width and length of the trapezium. This is because the angle in a triangle that creates the maximum area is 45°, I predict that this will be the angle that creates the largest area in the trapezium. If we look at the trapezium we see that it is basically a triangle divided into 2 and putting them on either side of a rectangle. Contrarily to the triangle and the rectangles area being 112.5cm2  I predict that the trapeziums area will be 112.52                              

                                sin

                                     cos

                                                        b

          As before doing every calculation would take to long so firstly I will change the angle 10° at a  time then when I have narrowed it down to less points I will change the angle 1° at a time so I can find the maximum area. When working out the area I need to consider the fact that there are 3 shapes 2 triangles and a rectangle.  I will work out the area for each shape then add it together and because the trapezium is symmetrical I only need to work out the area of 1 triangle then times it by 2.

              I then went on to investigate the maximum area of the trapezium. This investigation required very much more variation, in the form of both angles and lengths. I found the sides that produced the maximum area for this shape were all equal at 10cm each; the angle at which the sides fell was 120°. I noticed a quality shared by both the triangle with the maximum area and the hexagon with the maximum area; if the shapes were doubled they would produce regular polygons. The triangle produced a square and the trapezium produced a hexagon. The doubled triangle led me to notice another connection between investigated shapes.                       The doubled triangle was identical to the doubled rectangle, they were both 15x15cm. I remembered the fact that their areas were identical, this was because they were half the same rectangle.

The circle can be described, as having 1 side, in a true circle this is the case. A shape that has an unimaginable number of sides has the appearance of a circle but it is not. A circle must have a curved side. The polygon with sides can have millions of sides but the semi circle has the largest area as it has only 1 side. So from the start you could predict the semicircle to hold the most water as it has a curve and a curve will always create a larger area than a many sided shape even if it appears to look like a curve.            

Stephen Kyle