Scale Drawing Questions and Answers - Form 1 Topical Mathematics

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Questions

  1. Town X is 13.5km from town Y on a bearing of 028o. A matatu leaves y at 7:35 a.m towards a bearing of 080o. The matatu is at point Z due south of X at 8:55a.m.
    1. Calculate the average speed of the matatu from Y to Z
    2. If the matatu continues on the same bearing, calculate the distance it covers from Z when it is East of X.
  2. Three towns X, Y and Z are such that Y is 500km on a bearing of 315o from X. Z is on a bearing of 230o from X. given that the distance between Y and Z is 800km.
    1. using a scale of 1cm to represent 100km, draw a scale diagram to show the position of the Towns
    2. Find the bearing of;
      1. X from Z
      2. Z from Y
    3. Use the scale drawing to find the distance from X to Z
  3. Two aeroplanes S and R leave an airport at the same time. S flies on the bearing of 240o
    at 750Km/h while R flies due East at 600Km/hr.
    1.  
      1. Calculate the distance of each aeroplane after 30minutes
      2. Using a scale of 1cm to represent 50km make an accurate scale drawing to show the positions of the aeroplanes after 30minutes
    2.  
      1. Use the scale drawing to find the distance between the two aeroplanes after 30minutes
      2. If each aeroplane landed after 30minutes and S received a signal to join R in 45minutes. Find its speed.
    3. Determine the bearing of:
      1. S from R
      2. R from S
  4. The table below gives a field book showing the results of a survey of a section of a piece of land between A and E. All measurements are in metres.
    scale 4a
    1. Draw a sketch of the land.
    2. Calculate the area of this piece of land.
  5. Three towns A B and C are situated such that town A is 40km from B on a bearing of 280o. C is 60km from B on a bearing of 130o. Another town D is only 10km from C on a bearing of 210o.
    1. Drawing accurately and using a scale of 1cm to 10km find the distance from A to C and the bearing of A from C.
    2.  
      1. Distance of B from D
      2. Distance of A from D
      3. Bearing of A from D
      4. Bearing of C from D
  6. A train left Naivasha for Nakuru at 1000hours. It traveled at an average speed of 45km/h and reached Gilgil after 40minutes. It then covered the remaining 50km in 1½ hours. A second train left Nakuru for Naivasha at 1015 hours and arrived at Gilgil at the same time as the first train arrived at Nakuru.
    1. Using a scale of 1cm to represent 10minutes in the time axis and 1cm to represent 10km on the distance axis, draw on the same axes the graphs to show the movement of the two trains.
    2. use your graph to find;
      1. the distance between Naivasha and Nakuru
      2. the time at which the train met
    3. calculate the average speed, in km/h of the second train
  7. On a certain map, a road 20km long is represented by a line 4cm long. Calculate the area of a rectangular plot represented by dimensions 2.4cm by 1.5cm on this map – leaving your answer in hectares.
  8. A port B is on a bearings of 080o from a port A and at a distance of 95km. A submarine is stationed at a port D, which is on a bearing of 200o from A, and a distance of 124km from B. A ship leaves B and moves directly southwards to an island P, which is on a bearing of 140from A. The submarine at D on realizing that the ship was heading for the island P, decides to head straight for the island to intercept the ship.
    1. Using a scale of 1cm to represent 10km draw a diagram to show the positions of A,B,D, and P
    2. Hence;
      Determine;
      1. the distance from A to D
      2. the bearing of the submarine from the ship when the ship was setting off from B
      3. the bearing of the island P from D
      4. the distance the submarine had to cover to reach the island P
  9. Use a scale of 1cm represents 50km in these questions. Five towns A, B, C, D and E are situated such that A is 200 km from B on a bearing of 050° from E. C is 300 km from B on a bearing of 150° from B. D is 350km on a bearing of 240° from C. E is 200km from D and the bearing of D from E is 100°
    1. Draw the diagram representing the positions of the towns
    2. From the diagram, determine;
      1. The distance in km of A from E
      2. The bearing of D from B
  10. Four towns P, Q, R & S are such that P is 280 km North of R, S is 190 km from R on a bearing of 310o and Q is 240 km from P on a bearing of 105o.
    1. Using scale of 1 cm rep. 50 km, locate the four towns.
    2. Find;
      1. distance SQ.
      2. Bearing of S from Q.
      3. The shortest distance between P and side QR.
  11. Four ships are at sea such that a streamliner S is 150km on a bearing of 025° from a cargo ship C. A trawler T is 300km on a bearing of 145° from the cargo ship and a yacht Y is due West of C and on a bearing of 300° from T.
    1. Using a scale of 1cm= 50km, draw on accurate scale drawing showing the positions of S, C, T and Y
    2. By measurement from your scale drawing determine:
      1. The distance and bearing of Y from S
      2. The distance ST
      3. The distance YT
  12. A tea farm in Kakamega forest was surveyed and the results were recorded in the surveyors note book as shown below. The measurements are in meters.
    scale 12a
    Using a scale of 1:25, draw the map of the plot and hence calculate the area of the plot in Hectares.
  13. The information below shows the entries in a surveyor’s field book after a survey of a farm. XY = 280m is the baseline. All measurements are in metres.
    scale 13q
    1. Use a scale of 1cm represents 20m to draw the map of the farm
    2. Estimate the area of the farm in hectares
    3. If the point Y lies due north of X, find correct to 1 decimal place, the :
      1. Bearing of E from X
      2. Distance of E from X
  14. The measurements of a flower garden were recorded in a surveyor’s field book as shown.
    scale 14q
    Draw a sketch of the field and find its area. (Measurements are in m)
  15. A map has a scale 1:40,000:
    1. Calculate the distance between two points on the ground if the corresponding distance shown on the map is 3.25cm
    2. Calculate the area in the map of woodland which occupies 36ha on the ground
  16. Three scouts John, Peter and Samwel stand on three adjacent peaks of equal altitude on mountain range. The distance between John and Peter is 800metres and the bearing of Peter from John is 020o. The distance between John and Samwel is 1500metres, and the bearing of Samwel from John is 320o.
    1. Calculate the bearing of John from Peter
    2. Calculate:-
      1. the distance
      2. the bearing of Samwel from Peter
  17. The figure below represents a surveyor’s sketch of a plot of land. Calculate the area of the plot in square metres given that XY = 50m, XK = 20m, XM = 25m, XL = 35m, KA = 40m, MD = 38m and LB = YC = 60m.
    scale 17q
  18. Two boats P and Q are located 30km apart; P being due North of Q. An observer at P spots a ship whose bearing he finds as S 56oE from Q, the bearing of the same ship is 038o. Calculate the distance of the ship from Q to 2 decimal places
  19. A map is drawn to scale of 1:100,000. What area in km², is represented by a rectangle measuring 4.5cm by 5.4 cm
  20. Town B is 180 km on a bearing of 050o from town A. Another town C is on a bearing of 110from town A and on a bearing of 150o from town B. A fourth town D is 240 km on a bearing of 320o from A. Without using a scale drawing, calculate to the nearest kilometer.
    1. The distance AC
    2. The distance CD

Answers

  1.  
    1. YZ/Sin 28o = 13.5/sin 100o
      Duration of travel = 8:55am - 7:35am
      = 4/3
      Speed = 6.436 ÷ 4/3
      = 4.827 km/hr
    2. 13.5/Sin 10o = (6.436 + ZQ)/Sin 118o
       Sin 118o
      6.436 + ZQ = 13.5 x sin118o= 68.659
      ZQ= 68.659-6.436
      = 62.223

  2.  
    1.  
      scale 2a
    2.  
      1. 049 ± 1
      2. 190 ± 1
    3. 6.7 ± 0.1
      670 ± 10
  3.  
    1.  
      1. Distance covered by s
        = (750 x ½ )km = 375 km

        Distance covered by R
        = (600 x ½ ) km = 300 km
        scale 3a
    2.  
      1. Distance between the two aeroplanes
        = 12.5 x 50 = 625 ± 5 km
      2. Speed = (625/45 x 60) km/hr
        8331/3 km/h
    3.  
      1. Bearing of S from R = 225o
      2. The bearing of R from S = 72o
  4.  
    scale 4q
    Area A: ½ x 25 (33 + 21) = 675
    Area B: ½ x 40 (21 x 42) = 1260
    Area C: ½ x 30 x 42 = 630
    Area D: ½ x 25 x 40 = 500
    Area E: ½ x 5 (40 + 25) = 162.5
    Area F: ½ X 60 (25 + 36) = 1830
    Area G: ½ x 5 x 36 = 90 √
    = 5,147.5m2
  5.  
    1. A to C = 96 ± 1 km
      Bearing = 300o
    2.  
      1. 62 ± 1km
      2. 97 ± 1 km
      3. 304o
      4. 030o
  6.  
    1. Graph
    2.  
      1. 80 km
      2. 11.06a.m
    3. Average speed of the 2nd train
      Time taken = 80 ÷ 111/12
      = 80 x 12/23
      = 41.74km/h
  7. L.S.F = 4/2000000 = 1/500000
    A.S.F = 1/5 x 105 = 1/2.5 x 1011
    Area of rectangle = (2.4 x 1.5) cm2
    = 3.6cm2
    Actual area
    =3.6 x 2.5 x 1011 ha
       100 x 10000
    = 9 x 105
    = 900,000ha
  8.  
    1. Δ ABD correctly constructed
      Δ ABP
    2.  
      1. AD = 4.5 + 0.1cm
        Distance A + D
        = 4.5 X 10 = 45km
      2. Bearing of (i) from B
        = 241 + 1
      3. Bearing P from D
        = 123 = 2
      4. Dp = 12.9 + 0.2 am
        Distance D + P = 12.9 X 10
        = 129 km
  9.  

    1. scale 9a
    2.  
      1. 6.8 + 0.1cm
        Distance Ae = 340 + 5 km
      2. 180 + 18 = 198 + 2
  10.  
    1.  
      scale 10a
    2.  
      1. SP = 7.8 x 50 = 390 km ± 5 km
      2. S & Q = 255o ± 1o
      3. 4 x 50 = 200 km + 5 km
  11.  
    1. Scale = 50km
      Drawing accurately<NCE = 25o
      <NCT = 145o
      <NTY = 90o
      Lines drawn //
    2. By measurement:
      1. Distance SY = 6.9 x 50 = 345 ± 5km
        Bearing Y For S = 360o – 114 = 246±1o
      2. distance ST = 7.9 x 50 = 39.5 ± 5km
      3. distance YT = 9.8 x 50 = 490 ± 5km
  12. Area of A = ½ x 50 x 60 = 1500m2
    B = ½ x 70 x 60 = 2100m2
    C = ½ (60 +80) x 120=11050m2
    D = ½ x 80 x 80 = 3200m2
    F = ½ x 10 x 70 = 350m2
    Total area = 26600m2
    Ha = 26600/10000 = 2.66ha
  13.  


    1. scale 12q
    2.  
      Total area = area (1) + (2) + (3) + (4) +(5) + (6) + (7)
      Area (1)= ½ x 90 x 100 = 4500m2
      (2) = (100 + 105)10 = 10250m2
                     2
      (3) = ½ x 90 x 105 = 4725m2
      (4) = ½ x 50 x 110 = 2750m2
      (5) = ½ x (110 + 45)70 = 5425m2
      (6) = (45 + 95) 120 = 8400m2
      (7) = ½ x 40 x 95 = 1900m2
      Total area = 37,950m2
      In hectares = 37950/10000 ha = 3.795ha
    3.  
      1. bearing of E from x is 0.25 ± 1o
      2. Distance Ex = (12.8 x 0.1 x 20m) = 256 ± 2m
  14.  
    scale 14a
    Area A = ½ x 170 x 80 = 6800
    B = ½ x 80 x 80 = 3200
    C = ½ x 10 x 70 = 350
    D = ½ x 170 x 130 = 11050
    E = ½ x 70 x 60 = 2100
    Total = 23,500 m2
  15.  
    1. L.s.f/40,000 = 1
      1/40,000 = 3.25/x
      x = 130,000cm
    2. A.s.f = (1/40000)2
      (1/40000)2 = x/36000000
      x = 0.0225cm2
  16.  
    scale 16a
    1. bearing = 180 + 20 = 200o
    2. a2 = 1500 +
      a2 = b2 + c2 – 2bc cos A
      a2 = 15002 + 8002 – 2 x 1500 x 800cos 60
      = 2250000 + 640000 – 1200000
      = 1690000
      a = 1300m
    3. 1300/Sin 60 = 1500/sin c
      1300 sin c = 1500 sin 60
      Sin c = 1500 sin 60
                   1300
      = 0.9993
      c = 87.79o
      c = 87.80o
  17.  
    scale 17a
    A of Δ XYD = ½ x 50 x 38 = 950m2
    A of XBCY = ½ (50 + 15) 60
    = ½ x 65 x 60
    = 1950m2
    Total A = (950 + 1950)m2
    = 2900m2
  18. B1 for 86o
    30/Sin 86o = Q5/Sin 56o
    QS =
    30sin 56o
    Sin 86o
    = 24.93km
  19. 1cm for 100000cm
    1cm2 = (100000cm)2
    Area = 5.4 x 4.5 x 100000 cm2
    = 5.4 x 4.5 x 100000 x 100000Km2
       100000 x 100000
    = 24.3km2
  20.  

    1. scale 19a

      Use sine rule in ΔABC
      X/Sin 80o = 180/Sin 40o => x = 180 sin 80o/sin 40o
      = 275.8
      Hence AC = 276 km
    2. Use the cosine rule in Δ AD when < DAC = 150o
      y2 = 2402 + 2762 – 2 x 240 x 276 cos 150o
      = 576000 + 76180 – 132 480 (-cos 30o)
      = 133776 + 114731 = 248507
      y = √248507
      = 498.5
      Hence CD = 499 km
    3. Using sine rule in ΔABC we have
      BC    =  180
      Sin 60o sin 40o
    4. BC =
      180 sin 60
      Sin 40
      = 242.5
      = 243 km


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