PRE- CALCULUS

Application of Hyperbola in Real Life Situations

1. Precalculus Science, Technology, Engineering, and Mathematics Lesson 4.3 Applications of Hyperbolas in Real-life Situations

2. 2 Have you every wondered how we were able to track the locations of ships and aircrafts before we had navigation satellites (e.g. Global Positioning System or GPS)?

3. 3 Navigation systems such as the LORAN (long-range navigation) use radio signals to determine the exact location of a ship or aircraft.

4. 4 Knowledge of hyperbolas is used in this type of navigation system.

5. 5 What are some real-life applications of hyperbolas?

6. 6 Solve situational problems involving hyperbolas ( STEM_PC11AG-Ie-2 ).

7. 7 Recall the different parts and properties of a hyperbola. Solve word problems on hyperbolas.

8. 8 A hyperbola is defined as a set of points on a plane whose absolute difference between the distances from the foci, and , is constant.   Hyperbolas

9. 9 Parts and Properties of Hyperbolas Given arbitrary points and on the hyperbola, .   What is the distance ?  

10. 10 Parts and Properties of Hyperbolas  

11. 11 How do you determine the value of given the values of and ? How do you determine the value of given the values of and ?  

12. 12 Standard Equations of a Hyperbola Orientation of Principal Axis Equation Horizontal Vertical Orientation of Principal Axis Equation Horizontal Vertical

13. 13 Applications of Hyperbolas Hyperbolic Navigation Example: LORAN-C (long-range navigation)

14. 14 Applications of Hyperbolas Hyperbolic Navigation Two stations, and , transmit radio signals. The radio signals reach the ship at different times.  

15. 15 Applications of Hyperbolas Hyperbolic Navigation The location of the ship is somewhere on a hyperbola.  

16. 16 Applications of Hyperbolas Hyperbolic Navigation The difference between the distances of the two stations from the ship at point is .  

17. 17 Applications of Hyperbolas Cooling Towers Cooling towers use water from rivers and lakes to cool nuclear power plants.

18. 18 Applications of Hyperbolas Cooling Towers They protect aquatic life by ensuring that the water is returned to the environment at normal temperatures.

19. 19 Applications of Hyperbolas Cooling Towers Cooling towers are hyperboloid in shape. Why?

20. 20 Applications of Hyperbolas Cooling Towers able to withstand strong winds efficient in cooling economical

21. 21 Point is on a hyperbola. The distance of from the first focus is 6 units more than its distance from the second focus. What is the distance of the center to a vertex of the hyperbola?  

22. 22 3 units Point is on a hyperbola. The distance of from the first focus is 6 units more than its distance from the second focus. What is the distance of the center to a vertex of the hyperbola?  

23. 23 23 Point is on a hyperbola. The difference between the distances of from the first focus and the second focus is 20 units. What is the distance of the center to a vertex of the hyperbola?  

24. 24 What is the equation of a hyperbola whose center is at the origin, has a horizontal principal axis, the value of is , and the point is on the hyperbola?  

25. 25   What is the equation of a hyperbola whose center is at the origin, has a horizontal principal axis, the value of is , and the point is on the hyperbola?  

26. 26 26 What is the equation of a hyperbola whose center is at the origin, has a vertical principal axis, the value of is , and the point is on the hyperbola?  

27. 27 Point is on a hyperbola with a vertical principal axis and whose center is at the origin. The distance of from the first focus is 12 units more than its distance from the second focus . The distance between the two foci is 20 units. What is the equation of the hyperbola?  

28. 28   Point is on a hyperbola with a vertical principal axis and whose center is at the origin. The distance of from the first focus is 12 units more than its distance from the second focus . The distance between the two foci is 20 units. What is the equation of the hyperbola?  

29. 29 29 Point is on a hyperbola with a vertical principal axis and whose center is at the origin. The distance of from the first focus is 10 units more than its distance from the second focus . The distance between the two foci is 26 units. What is the equation of the hyperbola?  

30. 30 The sides of a cooling tower represent a hyperbola. The width of the slimmest part of the cooling tower is 50 meters. The length from this point to the top is 64 meters. The diameter of the top of the cooling tower is 60 meters. Find the equation of the hyperbola that represents the sides of the cooling tower. Set the middle of the slimmest part as the origin.

31. 31   The sides of a cooling tower represent a hyperbola. The width of the slimmest part of the cooling tower is 50 meters. The length from this point to the top is 64 meters. The diameter of the top of the cooling tower is 60 meters. Find the equation of the hyperbola that represents the sides of the cooling tower. Set the middle of the slimmest part as the origin.

32. 32 32 The sides of a cooling tower represent a hyperbola. The width of the slimmest part of the cooling tower is 54 meters. The length from this point to the top is 70 meters. The diameter of the top of the cooling tower is 68 meters. Find the equation of the hyperbola that represents the sides of the cooling tower. Set the middle of the slimmest part as the origin.

33. 33 Two stations and are along a straight coast such that station is 100 miles west of station . They both transmit radio signals at the speed of 186 000 miles per second. A ship sailing at sea is 50 miles from the coast. The radio signal from station arrives at the ship 0.0002 of a second earlier than the radio signal sent from station . Where is the ship? Set the midpoint of and as the origin.  

34. 34   Two stations and are along a straight coast such that station is 100 miles west of station . They both transmit radio signals at the speed of 186 000 miles per second. A ship sailing at sea is 50 miles from the coast. The radio signal from station arrives at the ship 0.0002 of a second earlier than the radio signal sent from station . Where is the ship? Set the midpoint of and as the origin.  

35. 35 35 Two stations and transmit radio signals such that station A is 200 miles west of station . Both stations sent radio signals with a speed of miles per microsecond to a plane traveling west. The signal from station reaches a plane 500 microseconds faster than the signal from station . If the plane is 80 miles north of the line from stations to , what is the location of the plane? Set the midpoint of and as the origin.  

36. 36 Let be a point on a hyperbola with center at the origin and foci and . Answer the following questions. Round off your answer to two decimal places.   If , what is ? If and , what is ? If and , what is ? If and , what is ? If and , what is ?  

37. 37 Solve the following problems. 1. A nuclear power plant has a cooling tower whose sides are in the shape of a hyperbola. The slimmest part of the tower is 58 meters. The length from this point to the top is 80 meters. The radius of the top of the cooling tower is 72 meters. What is the equation of the hyperbola that represents the sides of the cooling tower? Set the middle of the slimmest part as the origin.

38. 38 Solve the following problems. 2. The sides of a cooling tower represent a hyperbola. The width of the slimmest part of the cooling tower is 80 meters. The slimmest part of the tower is 90 meters from the ground. The diameter of the base of the tower is 130 meters. What is the equation of the hyperbola that represents the sides of the cooling tower? Set the middle of the slimmest part as the origin.

39. 39 Solve the following problems. 3. Two radio signaling stations and are 120 kilometers apart. The radio signal from the two stations has a speed of 300 000 kilometers per second. A ship at sea receives the signals such that the signal from station arrives 0.0002 seconds before the signal from station . What is the equation of the hyperbola where the ship is located? Set the midpoint of and as the origin.  

40. 40 A hyperbola is a set of points on a plane whose absolute difference between the distances from two fixed points and is constant. Given any point on a hyperbola with foci and , .  

41. 41 The distance from the center to a vertex of the hyperbola is , the distance from the center to an endpoint of the conjugate axis is , and the focal distance is . The relationship between the three distances is .  

42. 42 The standard forms of equation of a hyperbola whose center is at the origin are if the principal axis is horizontal , and if the principal axis is vertical .  

43. 43 In the real world, hyperbolas are used for navigation and the structure of cooling towers .

44. 44 To solve word problems on hyperbolas, follow these steps: Determine the standard form of equation of the hyperbola. Draw an illustration to be able to visualize the problem. Solve for the unknown equation or values.

45. 45 Concept Formula Description Relationship between the values , , and   where is the distance from the center to a vertex, is the distance from the center to an endpoint of the conjugate axis, and is the focal distance. Use this formula to find an unknown distance (e.g., focal distance) when the other values are known. Concept Formula Description Use this formula to find an unknown distance (e.g., focal distance) when the other values are known.

46. 46 Concept Formula Description Equation of a Hyperbola in Standard Form   where is the center; is the distance from the center to a vertex, and is the distance from the center to an endpoint of the conjugate axis. Use this formula to find the equation of a hyperbola given its center , , and if the transverse axis is horizontal . Concept Formula Description Equation of a Hyperbola in Standard Form  

47. 47 Concept Formula Description Equation of a Hyperbola in Standard Form   where is the center; is the distance from the center to a vertex, and is the distance from the center to an endpoint of the conjugate axis. Use this formula to find the equation of a hyperbola given its center , , and if the transverse axis is vertical . Concept Formula Description Equation of a Hyperbola in Standard Form  

48. 48 48 A nuclear power plant has a cooling tower that is hyperboloid in shape. The width of the slimmest part of the cooling tower is 60 meters and is 85 meters from the ground. The diameter of the base of the tower is 120 meters, while the diameter of the top is 96 meters. What is the height of the tower?

49. 49 Slides 2 and 3: Navigation boat Engineer Matusevich by Torin is licensed under CC BY-SA 3.0 via Wikimedia Commons .

50. Loran C Navigator by Morn the Gorn is licensed under CC BY-SA 3.0 via Wikimedia Commons . Slides 17 to 20 : Cooling towers of Dukovany Nuclear Power Plant in Dukovany, Třebíč District by Jiří Sedláček is licensed under CC BY-SA 4.0 via Wikimedia Commons .