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# The fencing problem.

Extracts from this document...

Introduction

Maths Coursework: The Fencing Problem

Varun Gupta

There is a need to make a fence that is 1000m long. The area inside the fence has to have the maximum area. I am investigating which shape would give this.

Triangles: Isosceles

To work out the area I need to know the height of the triangle. To work out the height I have to cut the triangle in half (which is why there is a line in the middle of the triangle). Then to work out the height I can use Pythagoras’ theorem:

a² + b² = c²

a² + 200² = 300²

a² = 300² - 200²

a² = 90,000 – 40,000

a² = 50,000

a = √50,000

height = 223.607m (3sf)

Now that I have calculated the height of the triangle I can now find the area of it.

Middle

316.228

47434.165

350

325.0

273.861

47925.724

400

300.0

223.607

44721.360

Looking at these results, it seems like as the base increases, the area also increases. However as the base is increasing, the height is decreasing. This makes the area decrease back again. The area is largest somewhere around the 300m-400m so I’m going to zoom in around that point and do exactly the same as I did in the table above except this time I am going to go up by 10m.

 BASE(m) EACH SIDE(m) HEIGHT(m) AREA(m²) 310 345.0 308.221 47774.209 320 340.0 300.000 48000.000 330 335.0 291.548 48105.353 340 330.0 282.843 48083.261 350 325.0 273.861 47925.724

Conclusion

Triangles: Scalene

Looking at this diagram, there is no need to draw out tables to find out whether or not a scalene triangle is bigger than an equilateral in terms of area. Logically, we know that no matter how high, or how far the scalene triangles go, they will never have the same area as an equilateral (provided that the perimeters for all of the triangles add up to 1000m) and the diagram above proves it all. In conclusion, my investigation has shown that out of all the three types of triangle, equilateral has the largest surface area.

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