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Geometrical Calculations in Various Scenarios

Explore geometrical calculations in real-life scenarios including boat travel, ramp safety, ladder angles, angle of elevation, and triathlon courses.

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Geometrical Calculations in Various Scenarios

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  1. 1. A boat leaves a port and sails 10 km north. It then turns and sails 24 km west. Draw a diagram to represent this information. (i)

  2. 1. A boat leaves a port and sails 10 km north. It then turns and sails 24 km east. At this point, what is the shortest distance between the boat and the port? (ii) x2 = 102 + 242 x2 = 100 + 576 x2 = 676 Shortest distance = 26 km

  3. 2. A ramp is 4 metres long and 304 millimetres high. In order for this ramp to be safe for wheelchair users the angle marked x must be 4° or less. Is this ramp safe for wheelchair users? 4 m = 400 cm = 4000 mm = 4·346° > 4° Therefore, the ramp is not safe.

  4. 3. For a ladder to be safe, it must be inclined at between 70° and 80° to the ground. The diagram shows a ladder resting against a wall. Is this ladder safe? (i) Label the sides of the triangle. Ladder (Opp) (Hyp) 5.4 m = 73·495° A 1.6 m Which is between 70º and 80º (Adj) Therefore, the ladder is safe.

  5. 3. For a ladder to be safe, it must be inclined at between 70° and 80° to the ground. The diagram shows a ladder resting against a wall. Find the length of the ladder, correct to the nearest centimetre. (ii) (Ladder)2 = (5·4)2 + (1·6)2 (Ladder)2 = 29·16 + 2·56 (Ladder)2 = 31·72 Ladder 5.4 m = 5·63 m = 563 cm 1.6 m

  6. 4. Freya stands at a point P, 7 m from the base of a 9 m tall vertical pole and measures the angle of elevation, A, from the ground to the top of the pole. Find the first angle of elevation, A, correct to the nearest metre. (i) Label the sides of the triangle. (Opp) (Hyp) (Adj)

  7. 4. Freya stands at a point P, 7 m from the base of a 9 m tall vertical pole and measures the angle of elevation, A, from the ground to the top of the pole. Freya then moves a further distance from the pole, to a point Q, and measures the angle of elevation from the ground to the top of the pole to be 31°.

  8. 4. Freya stands at a point P, 7 m from the base of a 9 m tall vertical pole and measures the angle of elevation, A, from the ground to the top of the pole. Find the distance Freya moved from P to Q. Give your answer to one decimal place. (ii) Label the sides of the triangle. (Hyp) (Opp) S (Adj)

  9. 4. Freya stands at a point P, 7 m from the base of a 9 m tall vertical pole and measures the angle of elevation, A, from the ground to the top of the pole. Find the distance Freya moved from P to Q. Give your answer to one decimal place. (ii) |PQ| = |SQ| − 7 = 14·978 − 7 = 7·978 |PQ| = 8·0 m S

  10. 5. A vertical pole stands on level ground. A cable joins the top of the pole to a point on the ground, which is 50 m from the base of the pole. The angle of depression from the top of the pole to the cable is 66° 25’. Find the height of the pole, correct to the nearest metre. (i) Alternate angles: Label the sides of the triangle. (Opp) (Hyp) Pole = 50 tan 66° 25′ = 114·53 = 115 m (Adj)

  11. 5. A vertical pole stands on level ground. A cable joins the top of the pole to a point on the ground, which is 50 m from the base of the pole. The angle of depression from the top of the pole to the cable is 66° 25’. Find the length of the cable, correct to the nearest centimetre. (ii) (Opp) (Hyp) (Adj)

  12. 6. From a point on the horizontal ground 12 m from the base of a building, the angle of elevation to the top of the building is 72° 24’. Draw a mathematical model to represent this information. (i)

  13. 6. From a point on the horizontal ground 12 m from the base of a building, the angle of elevation to the top of the building is 72° 24’. Find the height of the building, correct to two decimal places. (ii) x = 12 tan 72° 24′ x = 37·828 The height of the building = 37·83 m

  14. 7. A vertical mast [XY] stands on level ground. A straight wire joins Y, the top of the mast, to T, a point on the ground, which is 50 m from the bottom of the mast. If |YTX| = 56·31°, find |XY|, the height of the mast. (i) Label the sides of the triangle. (Hyp) (Opp) |XY| = 50 tan56·31° |XY| = 75 m (Adj)

  15. 7. A vertical mast [XY] stands on level ground. A straight wire joins Y, the top of the mast, to T, a point on the ground, which is 50 m from the bottom of the mast. A second straight wire joins Y to K, another point on the ground. If the length of this wire is 100 m, find |YKX|, correct to the nearest degree. (ii) Label the sides of the triangle. (Hyp) (Opp) (Adj) = 48·59° = 48°

  16. 8. In the diagram, ABCD represents the course in a triathlon. Competitors must swim the 9 km from A to B, then run the 12 km from B to C and cycle from C to D and back to A. |ADC| = 36·87°

  17. 8. In the diagram, ABCD represents the course in a triathlon. Competitors must swim the 9 km from A to B, then run the 12 km from B to C and cycle from C to D and back to A. |ADC| = 36·87° Find the distance from A to C (i) |AC|2 = 92 + 122 = 81 + 144 = 225 = 15 km

  18. 8. In the diagram, ABCD represents the course in a triathlon. Competitors must swim the 9 km from A to B, then run the 12 km from B to C and cycle from C to D and back to A. |ADC| = 36·87° Find the distance from C to D, correct to the nearest km. (ii) Label the sides of the triangle. (Opp) (Hyp) (Adj)

  19. 8. In the diagram, ABCD represents the course in a triathlon. Competitors must swim the 9 km from A to B, then run the 12 km from B to C and cycle from C to D and back to A. |ADC| = 36·87° Find the total length of the course. (iii) First find |AD| |AD|2 = 152 + 202 = 225 + 400 |AD|2 = 625 |AD| = 25 km

  20. 8. In the diagram, ABCD represents the course in a triathlon. Competitors must swim the 9 km from A to B, then run the 12 km from B to C and cycle from C to D and back to A. |ADC| = 36·87° Find the total length of the course. (iii) Total length of course = |AB| + |BC| + |CD| + |DA| = 9 + 12 + 20 + 25 25 km = 66 km 20 km

  21. 9. The angle of elevation of the top of a building, as viewed from a point A, 94 m from the base of the building is 27°. Find the height of the building, correct to the nearest metre. (i) Label the sides of the triangle. H = 94 tan 27◦ (Hyp) (Opp) = 47·89 Height of building = 48 m (Adj)

  22. 9. The angle of elevation of the top of a building, as viewed from a point A, 94 m from the base of the building is 27°. The bottom of a balloon is 72 m above the top of the building, as shown. Find the angle of elevation of the bottom of the balloon, as viewed from the point A. Give your answer correct to the nearest degree. (ii) Height of balloon = 72 + 48 = 120 m 48 m Angle of elevation = 52º

  23. 10. Two airplanes leave an airport, at noon. Plane S travels in the direction N 37° W at a speed of 410 km/hr. Plane T travels in the direction N 53° E at a speed of 280 km/hr. Find the distance each plane has travelled by 2:30 pm. (i) Distance of plane S = Speed  Time = 410  2·5 = 1025 km Distance of plane T = Speed  Time = 280  2·5 = 700 km

  24. 10. Two airplanes leave an airport, at noon. Plane S travels in the direction N 37° W at a speed of 410 km/hr. Plane T travels in the direction N 53° E at a speed of 280 km/hr. Find the distance between the planes at this time. Give your answer to the nearest kilometre. (ii) 37° + 53° = 90°. Use Pythagoras’ Theorem: 1025 km 700 km

  25. 11. A boat sails due east from the base A of a 30 m high lighthouse, [AD]. At the point B, the angle of depression of the boat from the top of the lighthouse is 68°. Ten seconds later the boat is at the point C and the angle of depression is now 33°. Find |BC|, the distance the boat has travelled in this time. (i) Step 1 First find |AB|

  26. 11. A boat sails due east from the base A of a 30 m high lighthouse, [AD]. At the point B, the angle of depression of the boat from the top of the lighthouse is 68°. Ten seconds later the boat is at the point C and the angle of depression is now 33°. Find |BC|, the distance the boat has travelled in this time. (i) Step 2 Find |AC|

  27. 11. A boat sails due east from the base A of a 30 m high lighthouse, [AD]. At the point B, the angle of depression of the boat from the top of the lighthouse is 68°. Ten seconds later the boat is at the point C and the angle of depression is now 33°. Find |BC|, the distance the boat has travelled in this time. (i) Step 2 cont’d

  28. 11. A boat sails due east from the base A of a 30 m high lighthouse, [AD]. At the point B, the angle of depression of the boat from the top of the lighthouse is 68°. Ten seconds later the boat is at the point C and the angle of depression is now 33°. Calculate the average speed at which the boat is sailing between B and C. Give your answer in metres per second, correct to one decimal place. (ii) 34·08 metres in 10 seconds 3·408 metres in 1 second 3·4 m/s

  29. 12. A cyclist is on a straight road heading directly north. At 10:30 am he passes a point A and observes a mobile phone mast on a bearing of N 40° E. At 11:20 am, he passes a point B and observes the same mobile phone mast on a bearing of S 70° E. The shortest distance between the road and the mobile phone mast is 4·7 km.

  30. 12. Draw a mathematical model to illustrate this information. (i) A cyclist is on a straight road heading directly north. At 10:30 am he passes a point A and observes a mobile phone mast on a bearing of N 40° E. Mast At 11:20 am, he passes a point B and observes the same mobile phone mast on a bearing of S 70° E. The shortest distance between the road and the mobile phone mast is 4·7 km.

  31. 12. Find the distance between the points A and B. (ii) Step 1 First find |AD| Label the sides of the triangle. (Opp) 5·6 km (Adj) (Hyp)

  32. 12. Find the distance between the points A and B. (ii) Step 2 Find |DB| Label the sides of the triangle. (Hyp) (Adj) 1·71 km (Opp) 5·6 km

  33. 12. Find the distance between the points A and B. Two Steps. (ii) Distance between A and B |AD| + |DB| = 5·6 + 1·71 1·71 km = 7·31 km 7·31 km 5·6 km

  34. 12. Hence, find the average speed of the cyclist, in km/hr, correct to one decimal place. (iii) Time taken = 11·20am – 10·30am = 50 mins Distance = 7·31 km 7·31 km = 8·772 = 8·8 km/hr

  35. Draw a mathematical model to illustrate this information. 13. (i) Two swimmers leave the same point, P, on the edge of a lake. Abby swims at a speed of 24 m/s in the direction N 20·32° W from P. Jed swims at a speed of 15∙5 m/s in the direction S 32·52° W from P. After 6 minutes Jed is due south of Abby.

  36. 13. Find the distance between Abby and Jed after 6 minutes. (ii) Two swimmers leave the same point, P, on the edge of a lake. Abby swims at a speed of 24 m/s in the direction N 20·32° W from P. Find the distance each swimmer has travelled in 6 minutes: Jed swims at a speed of 15∙5 m/s in the direction S 32·52° W from P. Abby: 6 minutes = 360 sec After 6 minutes Jed is due south of Abby. Distance = Speed  Time = 24  360 = 8,640 m Jed: Distance = Speed  Time = 15·5  360 = 5,580 m

  37. 13. Find the distance between Abby and Jed after 6 minutes. (ii) Step 1 Find |AB| (Hyp) (Opp) (Adj)

  38. 13. Find the distance between Abby and Jed after 6 minutes. (ii) Step 2 Find |BJ| (Adj) (Opp) (Hyp)

  39. 13. Find the distance between Abby and Jed after 6 minutes. (ii) Distance between Abby and Jed: 8102·3 m |AB| + |BJ| 8102·3 + 4705·08 12,807·38 m 12,807 m 4705·08 m

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