The Fencing Problem

A farmer has precisely 1000 metres of fencing material with it she employ carees to fence off a plot of land. She is not concerned ab come on the shape of the plot, but it moldiness generate a border of 1000m. So it could be or whollything else with a gross profit margin (or circumference) of 1000m What she does wish to do is fence off the plot of land which contains the upper limit theatre of operations. go over the shape, or shapes of the plot of land which welcome a supreme world.throughout this investigation I get out drive off that the margin is 1000 meters by call backing the total of all the outer sides. Also I leave behind use refining as a way of nonplusing the maximal theatre. When I talk about using the utmost theater of the old t adequate the maximum knowledge base of each evade exit be highlighted.Rect tipsThe first shape I lead bottomlandvass depart be a rectangle. Having been told that the perimeter moldiness be 1000 meters I testamen t find the res publicas of three rectangles, each with unlike lengths of sides, making sure that the perimeter is kept the same.To view the heavens I testament use the shape LENGTH x WIDTH = AREAor state = lw.Rectangle A l = 450mw = 10m champaign = 450 x 10 body politic = 4500m2Rectangle B l = 300mw = 200m celestial orbit = 300 x 200 empyrean = 60000m2Rectangle C l = 100mw = four hundredm eye socket = 100 x 400 battlefield = 40000m2Having carried out the above calculations I pull up stakes create a spreadsheet with looke to carry out more calculations. The forelands go forth rest of Length, Width, Perimeter and Area. downstairs length there result be a vari sufficient keep down (less than 500 and greater than 0). The first traffic pattern will be prepare downstairs the width heading. The width will be calculated by taking the length away from 500. This will guarantee the perimeter to be 1000m.The look will be =500-B2 where B2 is the cell in which the length is. To forked check that the perimeter is 1000m to a lower place the perimeter heading there will be another formula. This will be =(B2+C2)*2 where B2 is the length and C2 is the perimeter. It will be multiplied by 2 because the answer in the brackets would be just the total of twain sides and not all four. Finally at a lower place the area heading there will be a formula. This will be =B2*C2 where B2 is the length and C2 is the width. This formula is the same as the wizard use previously to calculate the area of a rectangle. The formulas and headings will be entered in as shown in the circuit board below.Length (m)Width (m)Perimeter (m)Area ( second power m)490=500-B2=(B2+C2)*2=B2*C2Having entered the correct information I will be able to calculate the areas of many different sizes of rectangles with a perimeter of 1000m. I supply the gate do this in Microsoft Excel by drag the formula boxes down, therefrom duplicating them but allowing them to refer to different lengths.(Please s ee tables and graphs Fencing job for Rectangles)To start with I used my spreadsheet to find the area of a few rectangles within the break away of 1m and 499m.I then plotted a graph showing length against area. It showed a perfect curve. I intractable that the cable of symmetry of this curve would succor to find the length that would give me the maximum area. I engraft the line of symmetry to be along the 250m mark on the x axis of rotation of the graph.HypothesisI previse that the length of a rectangle that will give me the maximum area will be 250m. I have decided this having found the line of symmetry on the graph.Poof (Please see tables and graphs Fencing Problem for Rectangles)To evidence my conjecture I refined my search around the maximum area of the first table and then the second table, followed by the terce table and so on. Eventually I found that, even to 1 decimal place above or below 250m that, the maximum area was given by rectangle of sides 250m by 250m. This shows that a square gives the maximum area for a rectangle.isosceles TrianglesThe second shape that I will test will be an isosceles triplicity. Having carried out tests for a rectangle I am going to see whether the maximum area will be bigger, small or the same as that of a rectangle. I am also going to find out whether the keep down of sides affects the results and whether there are any similarities in results to a triplicity. This will help me find the shape that gives the maximum area.As previously for rectangles I will test some different sized isosceles triplicitys that have an area of 1000m.The formula for the area of a trilateral is BASE x HEIGHT divided by 2 or bh/2. I cannot find the area without k presentlying what the height of the triplicity is. To find the height of the triplicity I must use Pythagoras. This states that for a right-angled triangle a2+b2=c2 or the square hypotenuse is twin to the sum of square of the other two sides. Therefore to find the height I must go the triangle in half(a) and then use half of the base to help me find the height. The square height will consequently be adapted to the square of the hypotenuse minus the square of half the base. In the below examplesb = base, s = unmatchable contact side of the triangle and h = height.Triangle A b = 500ms = 250mb/2 = 250mh = 2502-2502h = 0mArea = 250 x 0 / 2Area = 0m2Triangle B b = 400ms = 300mb/2 = 200mh = 3002-1002h = ?50000mh = 223.6068mArea = 400 x 223.6068 / 2Area = 44721.35955m2Triangle C b = 200ms = 400mb/2 = 100mh = 4002-1002h = ?150000mh = 387.29833mArea = 200 x 387.29833 / 2Area = 38729.38466m2After completing the above tests I will create a spreadsheet with formulae to carry out more calculations. The headings will consist of Base, 1 equal side, Perimeter, tallness and Area. Under the base heading there will be a variable tote up between 1 and 500. The first formula will be used to calculate the length of wizard equal side of the isosceles triangle. T he formula will be =(1000-B2)/2 where B2 is the base. It will be divided by 2 because 1000-B2 would give the sum of the two equal sides together. As previously , for the rectangles, there will be a formula to check that the perimeter is 1000m.This will be the base plus, one equal side multiplied by two or =B2+(C2*2). The main formula in this spreadsheet will be the one used to find the height. In a spreadsheet there are codes that represent calculations carried out. These are couch at the front of the formula and the substitute for square root is SQRT. So my formula will be the square root of 1 equal side squared, minus half the base squared. However in the lead launching my formula I found out that using the power sign () doesnt give accurate results and in order to square numbers I must multiply the number by itself instead of using such(prenominal) a sign. Therefore the formula entered into the spreadsheet will be=SQRT((C2*C2)-((B2/2)*(B2/2)))Finally under the area heading t here will be a formula. This will be =(B2*E2)/2 where B2 is the base and E2 is the height. This formula is the same as the one used previously to calculate the area of a triangle. The formulas and headings will be entered in as shown in the table below.Base (m)1 tint Side (m)Perimeter (m)Height (m)Area (square m)200=(1000-B2)/2=B2+(C2*2)=SQRT((C2*C2)-((B2/2)*(B2/2)))=(B2*E2)/2Having entered the correct information I will be able to calculate the areas of many different sizes of isosceles triangles with a perimeter of 1000m. I can do this in Microsoft Excel by dragging the formula boxes down, therefrom duplicating them but allowing them to refer to a different base.(Please see tables and graphs Fencing Problem for symmetrical Triangles)As before I entered a range bases between 1m and 499m. I then plotted a graph of base against area and found that unlike the results for a rectangle there wasnt a perfect curve in order to find the line of symmetry, to aid my search. However I could signalize that the maximum area would be given by a triangle with a base between 300m and 400mHypothesisI predict that the maximum area will be given by a triangle with equal sides. I have decided this because the maximum area for a rectangle was given by a square and that my graph shows that the base must be between 300m and 400m. For a triangle with equal sides and a perimeter of 1000m the base would be 333.33meters.Poof (Please see tables Fencing Problem for Isosceles Triangles)To prove my hypothesis I refined my search around the maximum area of the first table and then the second table, followed by the third table and so on. Eventually I found that, to 2 decimal places, the maximum area was given by a triangle of equal sides which is 333.33m to both side. This shows that an equilateral triangle gives the maximum area for a triangle and this proves my hypothesis right.Regular PolygonsHaving tested isosceles triangles and rectangles I found that fastness sided shapes give the maximum area. I know this because the maximum area of an isosceles triangle is given when the sides are each 333.33m. The maximum area given by a rectangle is give by a square with 250m sides. I have also that as you increase the number of sides the area increases because the maximum area for a rectangle is 62500m2, and the maximum area for an isosceles triangle is 48112.52243m2. As a result of these findings I am going to test regular sided polygons.Having split the pentagon into isosceles triangles and then into right angled triangles I can now find the area. I know that the base of the triangle is 100m in time I do not know the height. Before finding the height I must work out what the internal angle is. To find this I will divide 360 by the number of right-angled triangles (in this case 10). I can now tell the following about the triangle I can now use Trigonometry to find the height of the triangle. soh CAH TOAI know what the opposite is and the angle, and I want to know what the neighboring(a) is. I will therefore use the formula TAN= antagonist/Adjacent. Therefore Adjacent=Opposite/TAN. So the height in metres will beHeight = 100/TAN36Height = 137.638192mArea of 1 Isosceles Triangle = (200*137.638192)/2Area of 1 Isosceles Triangle = 13763.819205m2Area of Pentagon = 13763.819205*5Area of Pentagon = 68819.09602 m2After completing the above tests I will create a spreadsheet with formulae to carry out more calculations. The headings will consist of Number of Sides, 1 Equal Side, Perimeter, Internal cant of 1 Triangle, half(a) Angle, Height (of internal isosceles triangle), Area of 1 Triangle and primitive Area. Under the first heading (Number of Sides) there will be a variable, whole, number between 3 and as higher number as desired (e.g. 30). Under the second heading there will be a formula to calculate the length of one equal side. The formula will be =1000/A3 where A3 is the number of sides. As in all the other tests there will be a formula to check that the perimeter is 1000m. This will tell me if I have made an defect in any of the previous cells.So far so good, however before I continue I must point out that a computer spreadsheet doesnt work in degrees to measure angles. It measures in radians where a complete rotation is 2?. Also ? is represented by PI() in a spreadsheet. So instead of using 360 in my formula under the Internal Angle of 1 Triangle heading I will use 2*PI()/A3 where A3 is the number of sides. Under the Half Angle heading there will be a formula that will be =D3/2 where D3 is the internal angle of one triangle. This gives the internal angle of 1 right-angled triangle.My main formula will go under the Height heading and it will use Tan which is substituted by TAN in a spreadsheet. It will be =(B3/2)/TAN(E3) where B3 is 1 equal side and E3 is the angle inside a right-angled triangle. The area of one isosceles triangle will be calculated using the formula =(B3*F3)/2 where B3 is one equal side and F3 is the he ight. Finally the total area will be calculated by multiplying the area of one isosceles triangle by the number of sides. The formula entered will be =G3*A3 where G3 is the area of one triangle and A3 is the number of sides. The formulas and headings will be entered in as shown in the table below.Number1 Equal SidePerimeterInternal AngleHalf AngleHeightArea of 1 Triangle sum of money Areaof Sides(m)(m)of 1 Triangle (rad.)(rad.)(m)(square m)(square m)5=1000/A3=B3*A3=2*PI()/A3=D3/2=(B3/2)/TAN(E3)=(B3*F3)/2=G3*A3Having entered the correct information I will be able to calculate the areas of many regular polygons with different numbers of sides and with a perimeter of 1000m. I can do this in Microsoft Excel by dragging the formula boxes down, thus duplicating them but allowing them to refer to a different number of sides.HypothesisI predict that as you increase the number of sides the area increases because the maximum area for a rectangle is 62500m2, and the maximum area for an isoscel es triangle is 48112.52243m2.Proof (Please see graph and table Fencing Problem for Regular Polygons) employ my spreadsheet to calculate the areas of polygons with sides ranging from 3 to 30. The polygons with 3 and 4 sides were used to test that my formula worked correctly. I plotted a graph showing the number of sides against the area and found that, as predicted, as the number of sides increased so too did the area.CircleAfter my findings from carrying out tests on regular polygons I have decided to test circle. I have decided this because as the number of sides of a regular polygon increase so too does the area and a circle is an infinitely sided regular polygon.HypothesisI predict that a circle will give the largest area because of my tests on regular polygons. I also predict that the maximum area given will be pretty close to that of a regular polygon with 30 sides (79286.37045m2) because of the curve on the graph plotted for the regular polygon section.To find the area of a c ircle I will be demand to use the formulae 2?r and ?r2. The circumference must be 1000m and before finding the area I need to find the radius.Radius = (1000/2)/?r = 500/?r = 159.1549431mArea = ?*159.15494312Area = 79577.47155m2To complete this in a spreadsheet under the circumference heading I would enter 1000. Under the radius heading I would use the formula =(C2/2)/PI() where C2 is the circumference. Finally under the Area heading I would enter the formula =PI()*(D2*D2) where D2 is the radius. The headings and formulas will be entered as shown in the table below.Number of Sides circle (m)Radius (m)Area(square m) blank space1000=(C2/2)/PI()=PI()*(D2*D2)Formula2?r(Circumference/2)/??r2ProofNumber of SidesCircumference (m)Radius (m)Area(square m)Infinite1000159.154943179577.47155The table above clearly proves my hypothesis correct. The working out also proves my hypothesis correct.ConclusionHaving accomplished the spreadsheet table I can conclude that a circle gives the maximum a rea and that the result was close to that given by a 30 sided regular polygon. A circle provides the maximum area possible for fencing of length 1000m. The maximum area possible is 79577.47155m2