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My investigation starts with a simple question, this question is 'A farmer has 1000 meters of fencing and wants to know what shape with a 1000 meter perimeter would give the biggest area'. I am planning to start my investigation with the simplest shape or in other world the shapes with the least amount of sides. This means I shall be starting with circle because it has the least amount of or no sides some people would say and also the fact that there can only be one circle with a perimeter of 1000 meters.Prediction My prediction is that a rectangle or square will contain the largest area. The reasoning behind this prediction is that when ever you see a farmer's field in these days in is always a rectangle or a square so logical if the farmers today are using there land to its fullest a square or rectangle plot of land would contain the largest area.The circle
The diameter of a circle is 18.0886mThe radius of circle is 15.15441mThe area of a circle is 180.886mThe area of the circle is 180 to zero decimal placesNow I have worked out the area of a circle I will now move on to investigate rectangles and the one square which has a perimeter of 1000 meters. As there are a lot of rectangles with the perimeter of 1000 meters I'll be going around this a different way than the circle. I shall being using a spreadsheet on my p.c. because I only needed to work out one formula for each sum and instead of taking hours to work out all the sum it would seconds to doing it saving a lot of time. For example the formula for the area of a rectangle would be =AB. After I type the formula all I needed to do is to press enter and then drag the mouse cursor to the bottom right Corner of the section the section where I entered the formula and drag it down as far as I need to. length widtharea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he information in this table shows me that the rectangle with the highest area is in fact a square or other wise know as a regular rectangle. Triangles Now using the same technique which I used with the rectangles and one square I will try to work out the area, length and height of isosceles triangles using three different formulas. =(1000-A)/ I came about working out this formula by remembering that isosceles triangles have to sides the same. So I decided to go up in 10s starting at 10 and going up to 40, because if I when up to 500 the other to lengths would be 50. Meaning that the two sides would meet in the middle on the base line and if the middle line got any bigger the to other line would not actual meet. In maths terms in means 1000 take way the length and then divide it by two to gain the length. With the length worked out all that's left to work out the height because the area equals half the base times the height. To work out the height I need to half the triangle so it is a right angle triangle. From that I can then use Pythagoras's thermo to work out the height. Pythagoras's thermo is a +b =c. But in these triangles I only have a and c, meaning I will have to square a and then square c. Once that is done all I need to do is take a squared from c squared and then square root the answer. Now I know how to work out the height all I needed to do is turn it in a formula so the computer can understand it. The formula for the height is =SQRT((B)^-(A/)^) Which will be typed in and dragged down to the last base and length like I did with the area for the rectangle. Know with the height done and dusted all I needed to do is work out the area and then I'm finished with the triangles. Which is easy because the area of a triangle is half base times height. So the formula would be A divided by two in brackets then times by C.=(A/)C).BASEwidthHIEGTH AREA104544.747468474.877404048.87486488.74860485484.767857771.517864048047.581551.6604750475474.416411858.5416047046.04157614071.4787046546.68048168.8780460458.57565180.078045545.7656074.61656100450447.155560.6777110445441.58804487.481044045.88844615.66104540.11664757.55711404044.64068768.4848115045418.001174.7511604041.1056684.845170415406.010457.16180410400600010405.70077401.5740040087.84687.84610580.7886558.808810074.16578741158.1508567.4461445.6806408060.555175466.615150755.55064414.178607046.4101615450.17065.116445780.7780601.6647464.747060554.07044685.7006005016.77664744.164104508.0700147774.0850400048000051.54754748105.514008.847154808.6115057.861788475.77860064.575111476.56701554.5075747165.05801044.487446540.05110054.507884571.556640000.6067774471.55541051.10444487.067044000040004085187.088640.81614408017.0508088105.1177745075158.11885575.66846070141.415656.11470651.47448718781.50448480601004000405570.7106781174.11614 From my table I have found out that highest area lies between that two highlighted sections so I have decided to investigate further in to the triangles by making a new table with bases 1 to . Baselengthheightarea14.50.6888714810.005448.875448111.708.588.666554811.4501488.070584811.75.587.8148110.7116686.564148107.8787771.585.4804854810.75178184.604844808.410.58.7514801.4465From my table I have found out that highest area lies between that two highlighted sections so I have decided to investigate further in to the triangles by making a new table with bases .1 to ..Baselengthheightarea.1.4588.8771654811.4870..488.7058164811.5108..588.70400074811.5171.4.88.61784811.5155.5.588.507604811.5048.6.88.444104811.476.7.1588.5741714811.4504.8.188.7070614811.8085..0588.1864811.16From my table I have found out that highest area lies between that two highlighted sections so I have conclude that the highest area for a triangle would be a triangle with all sides of .. Which is also known as an equilateral triangle. My theory on regular shapesThrough my investigation I have created a theory that all the regular shapes for any shape will have the largest area out of all the other kinds of the same shape. I have come to this decision because of the tables I have made because the square was the highest and a square is basically a regular rectangle. As with the triangles as the equilateral triangle is a regular triangle and the highest triangle as well. So I came to the conclusion that if them to regular shapes have the highest area may all the regular shapes have the highest area. I also have another theory which is that as the number of sides goes up so does the area My theorem put to use SHAPESNumber of sides lengthsANGLESHIEGHTAREApentagon500717.6816881.060heptagon 6166.666666760144.75677168.78657 sided714.8571451.485714148.5674161.47847octagon81545150.88847675444.178nonagon111.11111114015.676447618.817decagon10100615.884176764.088411 sided110.0001.7777154.8065477401.871 sided18.0155.5011777751.05841 sided176.0767.6076156.044556780.7814 sided1471.4857145.7148571156.474506787.54715 sided1566.666666674156.8100778410.501816sided166.5.5157.104517855.175617sided1758.85411.1764705157.0447866.5118 sided1855.555555560157.55606178767.80061 sided15.61578518.47684157.701880578850.4050 sided05018157.84787781.8The results of my table match my prediction because as the sides rise so does the area. I have put this information in to a table because it makes it easier to read how the area goes up than just looking at all the sums and checking all 15 of them.My working for the lengths, heights, angles and areasI shall start with the pentagon and work my way down the table above.The lengths = 1000/5= 00. This is because a regular shape has all the same length sides. So to find out one side would be to find them all. So all I needed is divided the perimeter by the number of sides for ever shape. The angles= 60/5=7. The angles have basically the same principles as the length but for a different reason. This reason is that angles around a point all add up to 60 degrees. Like you can see in the diagram I have Highlighted the point in where the angles go round with a red circle. The pentagon has five triangles in it, there is a reason behind this reason, is that to find the area of a regular pentagon. I must first divide the regular pentagon in to 5 equal triangles so that once I know the base I can split one of the triangles in half so they both are right angle triangles.(as the triangles are isosceles triangles when the triangle is split both cantake one half of it and using trigonometry to work out the other side, which is also know as the height of the first triangle. I have identify the trigonometric function to use and it is, tan because opp = tan. So for the pentagon it would be7 adj tan6= 100m adj=100m tan6= 0.76545800560885854667574806 adjtan6100/tan6= 17.6810471175807058111Now with this figured out all I needed to do is multiply the answer which is above in red by 100 which is half the originals triangles base which would work out to look like this10017.6810471175807058111=.176.8104 (half base height) With this now known I shall multiply the answer again highlighted in red by the number of sides which in the case of the pentagon is five. So the area of the pentagon = 5176.8104= 6881.060. As I have proven that I now the way to work out the area of the pentagon I can now take the method I have used to work out the area of the pentagon and apply in it to the other shapes like the heptagon the shape with 6 sides. Simple by just changing the five in to a 6 for the length and area and then just carrying on the same processes that I used before so instead of adj=100/6tan in would be the adj=166.6666667/0tan. Then I would multiply the answer by half the base which would be 8.5 and I got the answer 7168.7865. With this knowledge I can know move on to finding the area of larger shapes, starting with the 50 sided shape and going up in 50 all the way to 1000. Using the same process which I have used above on all the shapes and again I will display my finding in a table so it is easier to read the results and discover a pattern . This table once again shows that my theorem about how the area goes up as the sides go up. With the to my knowledge I thought I would try and draw the shape with a 1000 sides and to the naked eye the 1000 sided shape look equally like a circle so this brought me to think that if in looked like a circle the circle would most likely have the largest are. I believe this because the circle has no Corners unlike the 1000 sided shape and the others, so with no corners this would give the circle the small but still bigger area. Just in check my theorem I am going to find the area of a million sided shape and if the circles area is still bigger I will be right.The length= 1000/1000000=0.001The angle= 60 /1000000=0.0006 Tan0.00018=Height =0.7Area=With this can I say that my prediction was wrong and that in fact the circle is the shape which posses the highest area out of all the shapes with 1000m perimeter. My investigation starts with a simple question, this question is 'A farmer has 1000 meters of fencing and wants to know what shape with a 1000 meter perimeter would give the biggest area'. I am planning to start my investigation with the simplest shape or in other world the shapes with the least amount of sides. This means I shall be starting with circle because it has the least amount of or no sides some people would say and also the fact that there can only be one circle with a perimeter of 1000 meters.Prediction My prediction is that a rectangle or square will contain the largest area. The reasoning behind this prediction is that when ever you see a farmer's field in these days in is always a rectangle or a square so logical if the farmers today are using there land to its fullest a square or rectangle plot of land would contain the largest area.The circle The diameter of a circle is 18.0886mThe radius of circle is 15.15441mThe area of a circle is 180.886mThe area of the circle is 180 to zero decimal placesNow I have worked out the area of a circle I will now move on to investigate rectangles and the one square which has a perimeter of 1000 meters. As there are a lot of rectangles with the perimeter of 1000 meters I'll be going around this a different way than the circle. I shall being using a spreadsheet on my p.c. because I only needed to work out one formula for each sum and instead of taking hours to work out all the sum it would seconds to doing it saving a lot of time. For example the formula for the area of a rectangle would be =AB. After I type the formula all I needed to do is to press enter and then drag the mouse cursor to the bottom right Corner of the section the section where I entered the formula and drag it down as far as I need to. length widtharea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he information in this table shows me that the rectangle with the highest area is in fact a square or other wise know as a regular rectangle. Triangles Now using the same technique which I used with the rectangles and one square I will try to work out the area, length and height of isosceles triangles using three different formulas. =(1000-A)/ I came about working out this formula by remembering that isosceles triangles have to sides the same. So I decided to go up in 10s starting at 10 and going up to 40, because if I when up to 500 the other to lengths would be 50. Meaning that the two sides would meet in the middle on the base line and if the middle line got any bigger the to other line would not actual meet. In maths terms in means 1000 take way the length and then divide it by two to gain the length. With the length worked out all that's left to work out the height because the area equals half the base times the height. To work out the height I need to half the triangle so it is a right angle triangle. From that I can then use Pythagoras's thermo to work out the height. Pythagoras's thermo is a +b =c. But in these triangles I only have a and c, meaning I will have to square a and then square c. Once that is done all I need to do is take a squared from c squared and then square root the answer. Now I know how to work out the height all I needed to do is turn it in a formula so the computer can understand it. The formula for the height is =SQRT((B)^-(A/)^) Which will be typed in and dragged down to the last base and length like I did with the area for the rectangle. Know with the height done and dusted all I needed to do is work out the area and then I'm finished with the triangles. Which is easy because the area of a triangle is half base times height. So the formula would be A divided by two in brackets then times by C.=(A/)C).BASEwidthHIEGTH AREA104544.747468474.877404048.87486488.74860485484.767857771.517864048047.581551.6604750475474.416411858.5416047046.04157614071.4787046546.68048168.8780460458.57565180.078045545.7656074.61656100450447.155560.6777110445441.58804487.481044045.88844615.66104540.11664757.55711404044.64068768.4848115045418.001174.7511604041.1056684.845170415406.010457.16180410400600010405.70077401.5740040087.84687.84610580.7886558.808810074.16578741158.1508567.4461445.6806408060.555175466.615150755.55064414.178607046.4101615450.17065.116445780.7780601.6647464.747060554.07044685.7006005016.77664744.164104508.0700147774.0850400048000051.54754748105.514008.847154808.6115057.861788475.77860064.575111476.56701554.5075747165.05801044.487446540.05110054.507884571.556640000.6067774471.55541051.10444487.067044000040004085187.088640.81614408017.0508088105.1177745075158.11885575.66846070141.415656.11470651.47448718781.50448480601004000405570.7106781174.11614 From my table I have found out that highest area lies between that two highlighted sections so I have decided to investigate further in to the triangles by making a new table with bases 1 to . Baselengthheightarea14.50.6888714810.005448.875448111.708.588.666554811.4501488.070584811.75.587.8148110.7116686.564148107.8787771.585.4804854810.75178184.604844808.410.58.7514801.4465From my table I have found out that highest area lies between that two highlighted sections so I have decided to investigate further in to the triangles by making a new table with bases .1 to ..Baselengthheightarea.1.4588.8771654811.4870..488.7058164811.5108..588.70400074811.5171.4.88.61784811.5155.5.588.507604811.5048.6.88.444104811.476.7.1588.5741714811.4504.8.188.7070614811.8085..0588.1864811.16From my table I have found out that highest area lies between that two highlighted sections so I have conclude that the highest area for a triangle would be a triangle with all sides of .. Which is also known as an equilateral triangle. My theory on regular shapesThrough my investigation I have created a theory that all the regular shapes for any shape will have the largest area out of all the other kinds of the same shape. I have come to this decision because of the tables I have made because the square was the highest and a square is basically a regular rectangle. As with the triangles as the equilateral triangle is a regular triangle and the highest triangle as well. So I came to the conclusion that if them to regular shapes have the highest area may all the regular shapes have the highest area. I also have another theory which is that as the number of sides goes up so does the area My theorem put to use SHAPESNumber of sides lengthsANGLESHIEGHTAREApentagon500717.6816881.060heptagon 6166.666666760144.75677168.78657 sided714.8571451.485714148.5674161.47847octagon81545150.88847675444.178nonagon111.11111114015.676447618.817decagon10100615.884176764.088411 sided110.0001.7777154.8065477401.871 sided18.0155.5011777751.05841 sided176.0767.6076156.044556780.7814 sided1471.4857145.7148571156.474506787.54715 sided1566.666666674156.8100778410.501816sided166.5.5157.104517855.175617sided1758.85411.1764705157.0447866.5118 sided1855.555555560157.55606178767.80061 sided15.61578518.47684157.701880578850.4050 sided05018157.84787781.8The results of my table match my prediction because as the sides rise so does the area. I have put this information in to a table because it makes it easier to read how the area goes up than just looking at all the sums and checking all 15 of them.My working for the lengths, heights, angles and areasI shall start with the pentagon and work my way down the table above.The lengths = 1000/5= 00. This is because a regular shape has all the same length sides. So to find out one side would be to find them all. So all I needed is divided the perimeter by the number of sides for ever shape. The angles= 60/5=7. The angles have basically the same principles as the length but for a different reason. This reason is that angles around a point all add up to 60 degrees. Like you can see in the diagram I have Highlighted the point in where the angles go round with a red circle. The pentagon has five triangles in it, there is a reason behind this reason, is that to find the area of a regular pentagon. I must first divide the regular pentagon in to 5 equal triangles so that once I know the base I can split one of the triangles in half so they both are right angle triangles.(as the triangles are isosceles triangles when the triangle is split both cantake one half of it and using trigonometry to work out the other side, which is also know as the height of the first triangle. I have identify the trigonometric function to use and it is, tan because opp = tan. So for the pentagon it would be7 adj tan6= 100m adj=100m tan6= 0.76545800560885854667574806 adjtan6100/tan6= 17.6810471175807058111Now with this figured out all I needed to do is multiply the answer which is above in red by 100 which is half the originals triangles base which would work out to look like this10017.6810471175807058111=.176.8104 (half base height) With this now known I shall multiply the answer again highlighted in red by the number of sides which in the case of the pentagon is five. So the area of the pentagon = 5176.8104= 6881.060. As I have proven that I now the way to work out the area of the pentagon I can now take the method I have used to work out the area of the pentagon and apply in it to the other shapes like the heptagon the shape with 6 sides. Simple by just changing the five in to a 6 for the length and area and then just carrying on the same processes that I used before so instead of adj=100/6tan in would be the adj=166.6666667/0tan. Then I would multiply the answer by half the base which would be 8.5 and I got the answer 7168.7865. With this knowledge I can know move on to finding the area of larger shapes, starting with the 50 sided shape and going up in 50 all the way to 1000. Using the same process which I have used above on all the shapes and again I will display my finding in a table so it is easier to read the results and discover a pattern . This table once again shows that my theorem about how the area goes up as the sides go up. With the to my knowledge I thought I would try and draw the shape with a 1000 sides and to the naked eye the 1000 sided shape look equally like a circle so this brought me to think that if in looked like a circle the circle would most likely have the largest are. I believe this because the circle has no Corners unlike the 1000 sided shape and the others, so with no corners this would give the circle the small but still bigger area. Just in check my theorem I am going to find the area of a million sided shape and if the circles area is still bigger I will be right.The length= 1000/1000000=0.001The angle= 60 /1000000=0.0006 Tan0.00018=Height =0.7Area=With this can I say that my prediction was wrong and that in fact the circle is the shape which posses the highest area out of all the shapes with 1000m perimeter. Please note that this sample paper on Women over 30 gained the vote in 1918 mainly because of women's contributions to the war effort. Do you agree is for your review only. 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