Maths Coursework: The Fencing Problem
A farmer has exactly 1000 metres of fencing, with it the farmer wishes to fence off a level plot of land. What the farmer does wish to do is fence off the plot of land which contains the maximum area.
Introduction
I have been given an investigation to examine which shape with a perimeter of 1000m will have the biggest area.
By using scale drawings, or a closed loop of string, I'm going to look at some possible figures for the plot of land. In each case I'm going to ensure that the perimeter is 1000m and obtain the enclosed area.
I'll then go onto further investigate the shape or shapes which have the maximum area and see whether there is a pattern within the concept of the tangent of an angle and the area of the shape.
Triangles
I'll begin by investigating the three - sided shapes. They consist of triangles of different shapes and sizes.
Equilateral Triangle
Area = 1/2 x 333? x 333? x Sin60?
Area = 48112.52m2 (2.d.p)
From this I can state that an equilateral triangle with all sides equal will have an area of 48112.52m2 (2 d.p.).
Table to show the area of 6 isosceles triangles.
Triangles
Shape
Side A
Side B
Side C
Area
290
290
420
42000.00
2
300
300
400
44721.35
3
310
310
380
46540.31
4
320
320
360
47623.52
5
330
330
340
48083.26
6
340
340
320
48000.00
From the table I've noticed that the closer the sides are to each other then the area is bigger.
I used Heron's formula to work out the area using the 3 sides:
VS(S-A) (S-B) (S-C) where S = (A+B+C) / 2
Since the equilateral triangle has all equal sides it should have the biggest area out of the triangles. In this case it's true with 48112.52 m2 (2 d.p.).
A farmer has exactly 1000 metres of fencing, with it the farmer wishes to fence off a level plot of land. What the farmer does wish to do is fence off the plot of land which contains the maximum area.
Introduction
I have been given an investigation to examine which shape with a perimeter of 1000m will have the biggest area.
By using scale drawings, or a closed loop of string, I'm going to look at some possible figures for the plot of land. In each case I'm going to ensure that the perimeter is 1000m and obtain the enclosed area.
I'll then go onto further investigate the shape or shapes which have the maximum area and see whether there is a pattern within the concept of the tangent of an angle and the area of the shape.
Triangles
I'll begin by investigating the three - sided shapes. They consist of triangles of different shapes and sizes.
Equilateral Triangle
Area = 1/2 x 333? x 333? x Sin60?
Area = 48112.52m2 (2.d.p)
From this I can state that an equilateral triangle with all sides equal will have an area of 48112.52m2 (2 d.p.).
Table to show the area of 6 isosceles triangles.
Triangles
Shape
Side A
Side B
Side C
Area
290
290
420
42000.00
2
300
300
400
44721.35
3
310
310
380
46540.31
4
320
320
360
47623.52
5
330
330
340
48083.26
6
340
340
320
48000.00
From the table I've noticed that the closer the sides are to each other then the area is bigger.
I used Heron's formula to work out the area using the 3 sides:
VS(S-A) (S-B) (S-C) where S = (A+B+C) / 2
Since the equilateral triangle has all equal sides it should have the biggest area out of the triangles. In this case it's true with 48112.52 m2 (2 d.p.).
