# Fencing Problem

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Introduction

GCSE Maths Coursework:- By Sarah Wolfe, 10Ba2 A farmer has exactly 1000 metres of fencing and wants t fence off plot of level land. I am going to come up with a range of shapes which has a perimeter or circumference of 1000 metres. For each shape I draw I will draw different size ones and find their areas. The reason for finding their areas is because the farmer wants to find a shape with the maximum area. To get started on this exercise, I am first of all going to come with a shape which I think is easiest to use; this will be a rectangle. 1) Perimeter = 450 + 50 + 450 + 50 = 1000m Area = 450 x 50 = 22,500m 2) Perimeter = 400 + 100 + 400 + 100 = 1000m Area = 400 x 100 = 40,000m 3) Perimeter = 385 + 115 + 385 +115 = 1000m Area = 385 x 115 = 44,275m 4) Perimeter = 375 + 125 + 375+ 125 = 1000m Area = 375 x 125 = 46,875m 5) Perimeter = 350 + 150 +350 +150 = 1000m Area = 350 x 150 = 52,500m 6) Perimeter = 325 +175+325 +175 = 1000m Area = 325 x 175 =56,875m Whilst finding the areas of these shapes, I'm finding that when I decrease the size of the length and increase the width, the area seems to get bigger. ...read more.

Middle

I predict that, like the square, the equilateral triangle will give the biggest area. 333.3 = 166.6 + h 111,111.1 = 277,7.7+ h h = 111,111.1- 277,7.7 h = 108,333.4 h = 108,333.4 h = 329.1m Area = b x h 2 = 333.3 x 329.1 2 Area = 54,850 m (to 1d.p) I am going to do another table, so I can see clearly the areas I have already found. Base (m) Height (m) Area (m ) 100 450 22,350 200 400 38,700 300 350 47,400 350 325 47,950 33.3 33.3 54,850 400 300 44,800 My predication was right and the reason I know this is because when you increase the length higher 33.3m, then the area starts to decrease, as you can see in my table. I am not going to do a graph, as I know it will give me the same sort of graph when I did one for the square; the graph will be symmetrical. I think that this will be the pattern for all of the shapes I do; the one which is equilateral will give the biggest area of that family. It seems to me that the shapes with the smaller sides, give the smallest area, so I am now going to move on to using shapes with more sides; the first one I will start with will be a pentagon. I am going to stick to finding the area of a regular pentagon because, before when finding the areas of the rectangles and triangles, my conclusion has been that the shape which is equilateral, gives the biggest area. ...read more.

Conclusion

Here are a set of instructions of how I did it. 1) From the centre of 2) Then, find the central the shape, split it up angle. (360 the into equal triangles. number of sides) Then label the length of each side. (1000m the number of sides) 4) Once you have found 3) Now find the area the area of that triangle, of one of the triangles; multiply the area by the to do this, you need to number of sides. This find the height first, now gives the final area using trigonometry. of that shape. I could also use a formula; e.g if I were to find the area of a 20 sided shape. I could use 'n' as the number of sides 1) Split it up into 20 triangles 2) Find central angle - 360 = ? 20 3) Find area of each triangle- need height (trigonometry) 4) 'n' (20) x area of one triangle = total area of shape Now I have got worked out the area of the all possible shapes, I am now going to put the shapes with their biggest area, into a table. Shape Area m (best area of shape) Rectangle (square) 62,500m Triangle 54,850m Pentagon 68,800m Hexagon 72,150m Octagon 75,450m 10 sided shape 20 sided shape Circle 79,572m I am now going to increase the number of sides a lot higher. I am not going to do the ones in between as I know already that the more sides I have, the area will be bigger. I am going to do a few more examples ...read more.

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