Math Coursework  The Fencing Problem
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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
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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 foursided 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. 
