• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Fencing Problem

Extracts from this document...

Introduction

Fencing Problem - Math's Coursework

A farmer has exactly 1000 metres of fencing and wants 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 1000 m. She wishes to fence off a plot of land that contains the maximum area. I am going to investigate which shape can provide her needs.

I am going to start by investigating the different rectangles; all that have a perimeter of 1000 meters. Below are 2 rectangles (not drawn to scale) showing how different shapes with the same perimeter can have different areas.

In a rectangle with a perimeter of 1000m, any 2 different length sides will add up to 500, because each side has an opposite with the same length. Therefore in a rectangle of 100m X 400m, there are two sides opposite each other that are 100m long and 2 sides next to them that are opposite each other that are 400m long. This means that you can work out the area if you only have the length of one side.

...read more.

Middle

30

14100

480

20

9600

490

10

4900

500

0

0

Using this table I can draw a graph of height against area. This is on the next sheet.

As you can see, the graph has formed a parabola. According to the table and the graph, the rectangle with a base of 250m has the greatest area. This shape is also called a square, or a regular quadrilateral. Because I only measured to the nearest 10m, I cannot tell whether the graph is true, and does not go up just to the sides of 250m. I will work out the results using 249m, 249.5 and 249.75.

Base (m)

Height (m)

Area (m2)

249

251

62499

249.5

250.5

62499.75

24975

250.25

6249993.75

250

250

62500

250.25

249.75

62499.9375

250.5

249.5

62499.75

251

249

62499

Using this table I can draw a graph of height against area. This is on the next sheet

All of these results fit into the graph line that I have, making my graph reliable. Also the equation that I used is a quadratic equation, and all quadratic equations form parabolas.

Now that I have found that a square has the greatest area of the quadirateral group, I am going to find the triangle with the largest area.

...read more.

Conclusion

As you can see from the graph, the line straightens out as the number of side's increases. Because I am increasing the sides by large amounts and they are not changing I am going to see what the result is for a circle. Circles have an infinite number of sides, so I cannot find the area using the equation for the other shapes. I can find out the area by using π. To work out the circumference of the circle the equation is πd. I can rearrange this so that diameter equals circumference/π. From that I can work out the area using the πr? equation.

DIAMETER = 1000 / π = 318.310

RADIUS = 318.310 / 2 = 159.155

AREA = π נ159.155? = 79577.472m?

My results:

Quadirateral: 62500m?

Triangle: 48107.689m?

Pentagon: 68819.096m?

Circle: 79577.472m?

From this I have concluded that a circle has the largest area when using a similar circumference. This means that the farmer should use a circle for her plot of land so that she can gain the maximum area.

...read more.

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. The Fencing Problem

    I will find the area by splitting the polygon in triangles. However I do not know the necessary information to find height from just this. I will need to split the triangles in half into equilateral triangles to find their height.

  2. The Fencing Problem

    90.5 399.98 1000 39998.48 100 400 91 399.94 1000 39993.91 100 400 91.5 399.86 1000 39986.29 100 400 92 399.76 1000 39975.63 Again, I have noticed the highest area has not varied. After the following diagrams, I will again display a graph corresponding to the areas in the table above; and I will conclude this stage of my investigation.

  1. Fencing Problem

    Rectangle 62499.99 1000 metres 249.1 x 250.1 2nd largest quadrilateral. The more narrowed down the closer dimension were to a square. Parallelogram (Trapezium) 39998.48 1000 metres On table of parallelograms. The smallest quadrilateral amongst the other 2. I stopped testing the values, as the area was nowhere near the area of a square.

  2. A farmer has exactly 1000m of fencing and wants to fence off a plot ...

    30 170 300 27658.63337 200 480 320 1000 500 20 180 300 23237.90008 200 490 310 1000 500 10 190 300 16881.94302 The maximum value in scalene is identical to an isosceles! This could mean that the more irregular shapes are the smaller the area, or this could be put

  1. Regeneration has had a positive impact on the Sutton Harbour area - its environment, ...

    But regeneration turned it around and developed it into a thriving leisure park for which thousands of people visit each year from around East Cornwall and West Devon. The old fish market is now a modern glassworks, which is a great boost to the local economy.

  2. Fencing problem.

    obviously noticed that the equilateral triangle possesses the most area out of all the other triangles where as a right angles triangle has the least area. Quadrilaterals During the second part of my coursework I shall be investigating different types of quadrilaterals.

  1. The Fencing Problem

    But we need to find the height of the triangle to find the Area. We know all the angles in the triangle so we can use trigonometry: Tan(72/2)= 100/H Rearranged this formula is: H=100/Tan46 H=137.64m Area=( X base X height Area=0.5 X 200 X 127.64 Area = 13763.82m2 But this

  2. Maths Fencing Coursework

    This made me realise that a square is a rectangle which is something I have learnt in this section which I didn't know before. Parallelograms I have now done squares and rectangles; I have to move on to quadrilaterals. I have chosen to do Parallelograms.

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