• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  • Level: GCSE
  • Subject: Maths
  • Word count: 1609

The Fencing Problem

Extracts from this document...

Introduction

Nick Murphy The Fencing Problem Mathematics GCSE In this piece of coursework I will be addressing the Fencing Problem. This is: A farmer has exactly 1000 meters of fencing; with it she wishes to fence off a plot of level land. She is not concerned about the shape of the plot, but it must have a perimeter of 1000m. What she does wish to do is fence off the plot of land, which contains the maximum area. Quadrilaterals I will start by looking at quadrilaterals. 50m 50 x 450 = 22500m� 45Om 400m 100m 100 x 400 = 40000m� 350m 150m 150 x 350 = 52500m� 200m 300m 200 x 300 = 60000m� 250m 250m 250 x 250 = 62500m� 250m As you can see I have drawn a large specturm of rectangles. I have noticed that the closer the size of the two measurements, the larger the area. Also, the closer the sizes are, the more like a square the shape becomes. In a rectangle, any two different length sides will add up to 500, because each side has an opposite with the same length. For example, in a 150 x 350 rectangle there are two sides opposite each other that are 150m long and two sides next to them that are opposite each other that are 350m long. ...read more.

Middle

+ h� 325m 325m 105625 - 30625 = 75000 h� = 75000 h = 273.9 350m 175m 1/2 base x height = 175 x 273.9 = 47932.5m� Hyp.� = h� + b� 333.3� = 166.7� + h� 111088.89 = 27788.89 + h� 111088.89 - 27788.89 = 83300 h� = 83300 h = 288.6 333.3m 333.3m 333.3m 333.3m 333.3m 166.7 1/2 base x height = 166.7 x 288.6 = 48109.6m� I have drawn here triangles with bases ranging from 50m to 350m going up in 50's. I have also drawn an equilateral triangle, as it is a regular triangle, and last time it was a regular quadrilateral that had the biggest area and it is the same in this case. Here is a base against area graph: As we can see the regular triangle has the largest area. Pentagon There are five sides to a pentagon, and it can be divided into five separate segments. The segments are isosceles triangles. We know the top angle is 72� by dividing 360 � 5, and therefore find the other two angles by 180 - 72 � 2 as the angles are equal. This equals 54� and because we can split an isosceles triangle into two right-angled triangles, I can use trigonometry to find the area. ...read more.

Conclusion

tan (90 - (180/n) = h 2 (1000/n) x h = Area of one segment 2 Area x n = Area of whole shape To prove this, here are some examples: Pentagon:- (1000/5) tan (90 - (180/5) = 138 2 (1000/5) x 138 = 13800 2 13800 x 5 = 69000m� Hexagon: - (1000/6) tan (90 - (180/6) = 144.3 2 (1000/6) x 144.3 = 12025 2 12025 x 6 = 72150m� Now that I have this equation, I can work out the area for an octagon and a nonagon. Octagon: (1000/8) tan (90 - (180/8) = 150.8 2 (1000/8) x 150.8 = 9425 2 9425 x 8 = 75400m� Nonagon: (1000/9) tan (90 - (180/9) = 152.6 2 (1000/9) x 152.6 = 8477.7 2 8477.7 x 9 = 76299.3m� Circle We have seen that as that as the number of sides increases, so does the area. Now I will find the area for a circle. Because a circle has an infinite number of sides, you cannot use any previous formula. To work out the area of a circle: 1000 = circumference pi 1000 = 318.3 � 2 = radius pi 318.3 � 2 = 159.15 pi x r� = Area pi x 159.15� = 79572.5 A = 79572.5m � In conclusion Here is a table and graph showing all my results for regular shapes: Therefore the circle has the greatest area with a 1000m circumference. ...read more.

The above preview is unformatted text

This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Fencing Problem essays

  1. Medicine and mathematics

    33 3.17127 34 1.90276 35 1.14166 36 0.68499 37 0.41100 38 0.24660 39 0.14796 40 0.08878 41 0.05327 42 0.03196 43 0.01918 44 0.01151 45 0.00690 46 0.00414 47 0.00249 48 0.00149 time dose 1 dose 2 dose 3 does 4 dose 5 0 300.0000 6 13.9968 313.9968 12 14.6498

  2. Fencing Problem

    Base Side A Side B Perimeter S Area (m�) 300 350 350 1000 500 47434.16 310 345 345 1000 500 47774.21 320 340 340 1000 500 48000.00 330 335 335 1000 500 48105.35 340 330 330 1000 500 48083.26 350 325 325 1000 500 47925.72 360 320 320 1000 500 47623.52 370 315 315 1000 500 47165.93 380 310

  1. Fencing problem.

    I shall now substitute this figure into the formula below: Area of the isosceles triangle = 1/2 � Base � Height Area of the isosceles triangle = 1/2 � 350m � 273.86m Area of the isosceles triangle = 47925.5m2 Right-angled triangle The third triangle that has to be explored is

  2. The Fencing Problem.

    Regular Polygons Having tested isosceles triangles and rectangles I found that regular sided shapes give the maximum area. I know this because the maximum area of an isosceles triangle is given when the sides are each 333.33m. The maximum area given by a rectangle is give by a square with 250m sides.

  1. Maths Coursework - The Fencing Problem

    I was able to come up with this formula.

  2. The Fencing Problem

    400 50 50 22360.68 100 448 452 1000 500 400 52 48 22342.78 100 446 454 1000 500 400 54 46 22289.01 100 444 456 1000 500 400 56 44 22199.10 100 442 458 1000 500 400 58 42 22072.61 100 440 460 1000 500 400 60 40 21908.90 Looking

  1. Fencing Problem

    For me to get the results as accurate as possible I will be using the hero's formula. A B C A+B+C are the three sides of the scalene triangle that add up to give 1000m. To find the area of this triangle I will need the Hero's formula that gives me the area.

  2. The Fencing Problem

    However, it increases at a decreasing rate. This is because with each area rise, the amount it rises by decreases each time. General formula Now that I know that as the amount of sides increases, the area increases, I can write a general formula on how to calculate any regular polygon with an n number of sides.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work