澳洲数学assignment格式.doc

Year 11 Math B assignmentLuran(Lori) ZhouMath AssignmentLuran(Lori) Zhou2012/2/24Year 11 Mathematic BTeacher: Miss Roach This assignment is about a cyclone is bearing down on the coast of China due east of Hong Kong and due south of Shang Hai, moving directly towards the town Yi Chang. The angles and the distances about the cyclone and towns were calculated by using Trigonometric functions and application.Q1. Choose a town between Hong Kong and Shang Hai and use the Internet site to find the latitude and the longitude of all the towns.CityThe latitudethe longitudeShang Hai31 13 48 N121 28 12 EHong Kong22 17 2 N114 9 EYi Chang30 42 N111 16 48 Ecyclone22 17 2 N121 28 12 EYi Chang was chosen as the town. The website was used to find the longitude and the latitude of all the towns and cities.The latitude and the longitude of Hong Kong: 22 17 2 N, 114 9 EThe latitude and the longitude of Shang Hai: 31 13 48 N, 121 28 12 EThe latitude and the longitude of Yi Chang: 30 42 N, 111 16 48 EThe latitude and the longitude of the cyclone: 22 17 2 N, 121 28 12 EQ2. Use the Internet Site http:/www.movable-type.co.uk/scripts/latlong.html to find and record the distance and initial bearing between the towns and cyclone.FromToDistanceInitial bearingHong KongShang Hai1231km0343347Shang Haicyclone994.8km1800000Hong Kongcyclone753.1km0883638Yi ChangHong Kong978.3km1622204Yi Changcyclone1379km1301900The website http:/www.movable-type.co.uk/scripts/latlong.html was used to find all the data above. AYi Chang1231km1379.0kmDCyclone753.1km994.8km978.3kmShang HaiCBHong KongQ3. Using measurements and sine and cosine rule to find all angles from the above diagram. The cosine rule formula was used to calculateA:cosA=b2+c2-a22bccosA=978.32+1379.02-753.122978.31379.0A=cos-1(978.32+1379.02-753.122978.31379.0)A=315150The sine rule formula was used to calculate ADB andABD:asin A=bsinB=csinCBDsin A=ABsinADB753.1sin315150=978.3sinADBADB=sin-1（978.3sin315150753.1）ADB=431744BDsin A=ADsinABD753.1sin315150=1379.0sinABDABD=sin-1（1379.0sin315150753.1）ABD=75934According to diagram, ABD is obtuseSo ABD=180-75934=1045026Verify the angles:ABD+ADB+A=1045026+431744+315150=1795936180In ABD, the three angles can be added into 180.The cosine rule formula was used to calculateA:cosA=b2+c2-a22bccosC=CD2+CB2-BD22CDCBcosC=994.82+12312-753.122994.81231C=cos-1(994.82+12312-753.122994.81231)C=37424The sine rule formula was used to calculate CBD andCDB:asin A=bsinB=csinCBDsin C=CDsinCBD 753.1sin37424=994.8sinCBDCBD=sin-1（994.8sin37424753.1）CBD=535257BDsin C=CBsinCDB753.1sin37424=1231sinCDBCDB=sin-1（1231sin37424753.1）CDB=882422Verify the angles:CDB+CBD+C=882422+535257+37424=1795923180In CBD，the three angles can be added to 180.Q4. Find the area formed by the lines joining these towns from the above diagram.The surface area formula was used:SA =12bcsinASA ABD=12ABADsinBAD =12978.31379.0sin315150 =356090.17km2SA CBD=12CBCDsinBCD =12994.81231sin37424 =374447.03km2The area of triangle ABD is 356090.17km，the area of triangle CBD is 374447.03Km.Shang HaiCEQ5. Verify the bearing using trigonometry and the latitude and longitude values from the diagram below.FAYi Chang1231km1379.0km978.3km994.8km753.1kmDCycloneBHong KongFAD=ADB=431744EAB=EAF+FAD+BAD=90+431744+315150 =165934The initial bearing of Yi Chang to Shang Hai calculated by using latitude and longitude is 1622204, and the trigonometry calculation is 165934Q6. Write any assumptions, strengths and/or limitations relating to the calculations above.Assumptions: 1. It was assumed to happen in 2 dimension not 3 dimensions. The earth should be a sphere, but it consider as a flat surface. The distance between the towns and the cyclone is a straight line not curved line.2. Its was assumed that the centre of the towns, cities and cyclone are joined together.3. The calculations which were provided by the websites were assumed to be accurate.Limitations:1. The calculations of the questions were not very accurate, they were approximate value. For example, most of the angles were approximate to the neatest seconds.2. The distance between each town and city from the websites may not be accurate.Q7. Find the distance from the weather to each town when the angle to the other town has reduced by a half.D1231km1379.0km978.3km994.8km753.1kmEDCycloneAYi ChangBHong KongShang HaiCIn the diagram above, the cyclone was assumed to move directly to the town Yi Chang on AD. And when the cyclone moved to D, the angle BCD was divided in half by CD. The cyclone D and each town are joined together, A, E, D, D are on the same line.According to Q2,A=315150, ADB=431744, ABD=1045026, C=37424, CBD=535257, CDB=882422CD divided BDC in halfBCD=DCD=12BDC=1237424=18512CDD=CDB-ADB=882422-431744=45638In CDD, DCD=18512, CDD=45638, CD=994.8kmCDD=180-DCD-CDD=180-18512-45638=116220The sine rule formula was used:asin A=bsinB=csinCCDsin CDD=CDsinCDDCDsin45638=753.1sin116220CD=753.1sin45638sin116220CD=784.41kmDDsin DCD=CDsinCDDDDsin18512=753.1sin116220DD=753.1sin18512sin116220DD=270.81kmAD=AD-DD=1379-270.81=1108.19kmIn CBD, DCD=18512, CD=784.41Km, CB=1231kmThe cosine rule formula was useda2=b2+c2-2bccosABD2=CB2+CD2-2CBCDcosBCDBD2=12312+784.412-21231784.41cos18512=303024.3686BD=550.48kmThe distance from the weather to Hong Kong is 550.48km; from the weather to Yi Chang is 1108.19km and from the weather to Shang Hai is 784.41km.Q8. Write any assumptions or approximations that are relevant to the calculations.Assumptions: 1. The cyclone was assumed to move straight on a line to the town.2. The centre of the town, cities and cyclone were joined together.3. It h