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.
Now that I have found that a square has the greatest area of the rectangles group, I am going to find the triangle with the largest area. I am only going to use isosceles triangles because if I know the base I can work out the other 2 lengths because they are the same. If the base is 200m long then I can subtract that from 1000 and divide it by two. This means that I can say that:
Side = (1000 – 200) / 2 = 400
To work out the area I need to know the height of the triangle. Tow ork out the height I can use Pythagoras’ Theorem. Below is the formula and area when using a base of 200m.
H² = h² - a²
H² = 400² - 100²
H² = 160000 - 10000
H² = 150000
H = 387.298
½ × 200 × 387.298 = 38729.833m
Below is a table of results for isosceles triangles from the base with 10m to a base with 500m.
Because the last two shapes have had the largest areas when they are regular, I am going to use regular shapes from now on.
The next shape that I am going to investigate is the pentagon.
Because there area 5 sides, I can divide it up into 5 segments. Each segment is an isosceles triangle with the top angle being 72º. This is because it is a fifth of 360º. This means that I can work out both the other angles by subtracting 72 from 180 and dividing the answer by 2. This gives 54º each. Because every isosceles triangle can be split into 2 right-angled triangles, I can work out the area of the triangle, using trigonometry. I also know that each side is 200m long, so the base of the triangle will be 100m.
Using SOH CAH TOA I can work out that I need to use Tangent.
H = 100 tan54 = 137.638
O = 100
T = tan 36
This gives me the length of H so I can work out the area.
Area = ½ × b × H = ½ × 100 × 137.638 = 6881.910
I now have the area of half of one of the segments, so I simply multiply that number by 10 and get the area of the shape.
Area = 6881.910 × 10 = 68819.096m²
All of the results that I have got so far have shown that as the number of sides increase, so to does the area. Using a spreadsheet and formula I have created a table that shows my prediction is right. This is show on the next page.
The formulae for the spreadsheet are:
To work out the base of a polygon you divide the perimeter of the polygon by the number of side (n)
To put this equation in to a spreadsheet, you must type the following:
=(1000/A3)
To work out the height of the triangle on a polygon, the equation is:
To put this equation in to a spreadsheet, you must type the following:
=(500/A3)/TAN(3.14/A3)
The equation to work out the area of the triangle is:
To put this equation in to a spreadsheet, you must type the following:
=(B3*C3)/2
To work out the area of the polygon, the equation is:
To put this equation in to a spreadsheet, you must type the following:
=(A3*D3)
From the method that I used to find the area for the pentagon I can work out a formula using N as the number of sides. To find the length of the base segment I would divide 1000 by the number of sides. Also on the next page is a graph showing the number of sides against area.
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 cir le the equation is πd. I can rearrange this so that diameter equals circumference/π. From that I can work out the area using the πr² equation.
DIAMETER = 1000 / π = 318.310
RADIUS = 318.310 / 2 = 159.155
AREA = π × 159.155² = 79577.472m²
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
What impression did William Woodruff give of the visitors from America? How does the author use language to show the different reaction s of the family to the visitors?
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