Math Coursework - The Fencing Problem


Length: 909 words (2.6 double-spaced pages)
Rating: Excellent
Open Document

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Text Preview

More ↓

Continue reading...

Open Document

The Fencing Problem

A farmer has 1000m of fencing and wants to fence off a plot of level
land.

She is not concerned about the shape of plot, but it must have a
perimeter of 1000m. So it could be:

[IMAGE]

Or anything else with a perimeter (or circumference) of 1000m.

She wishes to fence of the plot of land with the polygon with the
biggest area.

To find this I will find whether irregular shapes are larger than
regular ones or visa versa. To do this I will find the area of
irregular triangles and a regular triangle, irregular quadrilaterals
and a regular square, this will prove whether irregular polygons are
larger that regular polygons.



Area of an isosceles irregular triangle:
========================================

(Note: I found there is not a right angle triangle with the perimeter
of exactly 1000m, the closest I got to it is on the results table
below.)

To find the area of an isosceles triangle I will need to use the
formula 1/2base*height. But I will first need to find the height. To
do this I will use Pythagoras theorem which is a2 + b2 = h2.

[IMAGE]

[IMAGE]




First I will half the triangle so I get a right angle triangle with
the base as 100m and the hypotenuse as 400m. Now I will find the
height:

a2 + b2= h2

a2 + 1002 = 4002

a2 = 4002 - 1002

a2 = 160000 - 10000

a2 = 150000

a = 387.298m

Now I will find the area:

100*387.298 = 3872.983m2

My table shows the areas of other irregular triangles, but to prove
that regular shapes have a larger area I will show the area of a
regular triangle:

Area of a regular triangle:

Tan30= 166.6666667/x

X= 166.666667/Tan30

X= 288.675m

288.675*166.6666667

= 48112.5224m2

This shows clearly that the regular triangle's area is larger than the

Need Writing Help?

Get feedback on grammar, clarity, concision and logic instantly.

Check your paper »

How to Cite this Page

MLA Citation:
"Math Coursework - The Fencing Problem." 123HelpMe.com. 22 Feb 2018
    <http://www.123HelpMe.com/view.asp?id=120592>.
Title Length Color Rating  
Math Coursework - The Fencing Problem Essay - The Fencing Problem Introduction A farmer has exactly 1000 metres of fencing and wants to use it to fence a plot of level land. The farmer was not interested in any specific shape of fencing but demanded that the understated two criteria must be met: · The perimeter remains fixed at 1000 metres · It must fence the maximum area of land Different shapes of fence with the same perimeter can cover different areas. The difficulty is finding out which shape would cover the maximum area of land using the fencing with a fixed perimeter....   [tags: Math Coursework Mathematics] 657 words
(1.9 pages)
Strong Essays [preview]
Math Coursework - The Fencing Problem Essay - The Fencing Problem A farmer has 1000m of fencing and wants to fence off a plot of level land. She is not concerned about the shape of plot, but it must have a perimeter of 1000m. So it could be: [IMAGE] Or anything else with a perimeter (or circumference) of 1000m. She wishes to fence of the plot of land with the polygon with the biggest area. To find this I will find whether irregular shapes are larger than regular ones or visa versa. To do this I will find the area of irregular triangles and a regular triangle, irregular quadrilaterals and a regular square, this will prove whether irregular polygons are larger that regular polygons....   [tags: Math Coursework Mathematics] 909 words
(2.6 pages)
Strong Essays [preview]
Math Coursework - The Fencing Problem Essay - The Fencing Problem Aim - to investigate which geometrical enclosed shape would give the largest area when given a set perimeter. In the following shapes I will use a perimeter of 1000m. I will start with the simplest polygon, a triangle. Since in a triangle there are 3 variables i.e. three sides which can be different. There is no way in linking all three together, by this I mean if one side is 200m then the other sides can be a range of things. I am going to fix a base and then draw numerous triangles off this base....   [tags: Math Coursework Mathematics] 1214 words
(3.5 pages)
Strong Essays [preview]
Essay on The Fencing Problem - Mathematics - The Fencing Problem Introduction ============ I have been given 1000 meters of fencing and my aim is to find out the maximum area inside. ====================================================================== Prediction ---------- I would predict that the more sides the shape has, then possibly the bigger the area it will have, although I have nothing to base this on, it will be what I am about to investigate. Shapes: I am going to start with the rectangle, I think this is a good starting block because I am able to vary the widths and lengths to see which has the bigger area....   [tags: Math Coursework Mathematics] 890 words
(2.5 pages)
Strong Essays [preview]
Essay about Mathematics - The Fencing Problem - Fencing Problem A farmer has exactly 1000 meters of fencing and wants to fence of 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 of a plot of land that contains the maximum area. I am going to investigate which shape is best for this and why. 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....   [tags: Math Coursework Mathematics] 1293 words
(3.7 pages)
Strong Essays [preview]
Essay about Introduction to Solve Math Problems Deductive Reasoning - Introduction to solve math problems deductive reasoning Deductive reasoning is one of the two essential forms of suitable reasoning. The reasoning constructs or evaluates deductive reasoning. While deductive reasoning argues from the general to exacting , similarly inductive reasoning argues from the specific to a general instance. Deductive arguments may be valid or invalid,and sound or unsound, but that are not true or false. Whenever we turn up for the conclusion using facts, definitions, rule, or properties, then it is so called Deductive Reasoning....   [tags: Math] 508 words
(1.5 pages)
Good Essays [preview]
The Fencing Problem Essay - The Fencing Problem Introduction I am going to investigate different a range of different sized shapes made out of exactly 1000 meters of fencing. I am investigating these to see which one has the biggest area so a Farmer can fence her plot of land. The farmer isnÂ’t concerned about the shape of the plot, but it must have a perimeter of 1000 meters, however she wishes to fence off the plot of land in the shape with the maximum area. Rectangles I am going to look at different size rectangles to find which one has the biggest area....   [tags: Papers] 2291 words
(6.5 pages)
Strong Essays [preview]
Essay on The Fencing Problem - The Fencing Problem 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: Scalene [IMAGE] The diagram above is not to scale. Instead of having the perimeter to 1000m, only in this diagram, I have made the perimeters of the shape to 10, only to make this part of the investigation easier to understand. We know that the base of all the shapes is 2. The lengths for the equilateral triangle are 4 on each side....   [tags: Papers] 2933 words
(8.4 pages)
Strong Essays [preview]
Math Fencing Project Essays - Math Fencing Project I have to find the maximum area for a given perimeter (1000m) in this project. I am going to start examining the rectangle because it is by far the easiest shape to work with and is used lots in places (most things use rectangles for design- basic cube .etc). To start with what type of rectangle gives the best result. A regular square or an irregular oblong. I start by having 4 individual squares. [IMAGE] [IMAGE] [IMAGE] [IMAGE] [IMAGE][IMAGE] Goes to [IMAGE] [IMAGE] Regular square irregular oblong Now look at how many sides are exposed on each shape- Ã¥ sides of each cube internal1 Ã¥ sides of each cube internal2...   [tags: Papers] 1125 words
(3.2 pages)
Strong Essays [preview]
Fencing Problem Essay - Fencing Problem Introduction: A farmer has 1000 metres of fencing, and he wants to be able to get the maximum amount of space that he can with the 1000 metres. By using formulae that we already know, we can find out what shape can give the most area. I will test each shape to find the maximum area I can, then I can eventually use the shape that can create the largest area with 1000 metres of fencing. First I will look at Rectangles, then all 3 types of triangles, Pentagons, hexagons and finally circles, and by the end I will have found out which is the suitable shape to have the fence in....   [tags: Papers] 1176 words
(3.4 pages)
Strong Essays [preview]

Related Searches




isosceles triangle's area. To back my point up I will look at the
irregular quadrilateral's area and a regular square's area.

The formula to find a four-sided shape's area is b*h.

Irregular quadrilateral:

[IMAGE]




[IMAGE][IMAGE]Base= 450m

Height= 50m

50*450= 22500m2

Regular quadrilateral:

[IMAGE]




All sides= 250m

250*250= 62500m2

To further back up my prediction that regular polygons have a larger
area I have made a table (below) that clearly shows that the polygons
I have researched have a smaller area. So I will see what regular
polygon has the largest area. I also found that there are the same
amount of isosceles triangles in the regular polygon as there are
sides:

[IMAGE]

[IMAGE]




When finding out whether regular polygons have a larger area than
irregular polygons I found that shapes with a larger amount of sides
have a larger area. E.g. regular triangle area= 48112.5224m2, regular
quadrilateral area= 62500m2. So I decided to find the polygon with the
largest area. To do this more efficiently I devised a formula in excel
and by hand which make the process faster. There are two separate
formulas because excel works with radians so I needed to adapt the
formula so it could work in excel. Here are the two formulae:

[IMAGE]

In excel: (aaa= this will change for each shape, it is a cell
reference and refers to the amount of sides in the polygon)

=(500/A3)/(TAN(PI()/A3))*500




This is what the equation means:

[IMAGE]The 500/n calculates the height of the right angle triangle
within a polygon.

[IMAGE] Tan 180/n

[IMAGE]




[IMAGE][IMAGE]The x 500 is the simplified version of my first
equation: 1000 x n 500 = x 500

[IMAGE][IMAGE]

N

[IMAGE][IMAGE] 2 1

This finds the overall area of the regular polygon and also simplifies
out to what is now x 500.

The excel formula is the same, but excel works with radians so I
needed to change the equation so that I works with radians. Therefore
instead of Tan 180/n, it is now tan p/n (cell reference). This is
because p= 180°.

So if I wanted to find the area of a decagon I would do the following
by hand:

500/10 x 500

[IMAGE]


Tan 180/10

= 50/ Tan 18 x 500

=76942.088m2

Because I can put my formula in excel I can see the area from a
variety of different polygons. Here are my results:

Sides in polygon

Area of polygon (m2)

Sides in polygon

Area of polygon (m2)

3

48112.5224

28

79243.2632

4

62500.0000

29

79265.9324

5

68819.0960

30

79286.3705

6

72168.7836

31

79304.8609

7

74161.4784

32

79321.6437

8

75444.1738

33

79336.9227

9

76318.8172

34

79350.8725

10

76942.0884

35

79363.6429

11

77401.9827

36

79375.3632

12

77751.0585

100

79551.2899

13

78022.2978

200

79570.9265

14

78237.2548

300

79574.5626

15

78410.5018

400

79575.8353

16

78552.1796

500

79576.4243

17

78669.5221

1000

79577.2097

18

78767.8031

2000

79577.4061

19

78850.9402

3000

79577.4425

20

78921.8939

4000

79577.4552

21

78982.9345

5000

79577.4611

22

79035.8270

23

79081.9600

24

79122.4387

25

79158.1509

26

79189.8169

27

79218.0258

From these results I plotted a graph (separate sheet). From the graph
you can see that the more sides there are in a shape the larger the
area. I decided to test one more shape and that is a circle. A circle
with a circumference of 1000m has the largest area. From this I can
say that the circle is the shape with the most amount of sides and is
also a regular polygon. But also the circle has a infinite amount of
sides which also makes it the shape with the largest area.

Therefore according to all my calculations I can safely say that the
fence should be made into a regular circle shape with a perimeter of
1000m.


Return to 123HelpMe.com