- 1. Capstone Project Science, Technology, Engineering, and Mathematics 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. Learning Competency At the end of the lesson, you should be able to do the following: 6 Solve situational problems involving hyperbolas (STEM_PC11AG-Ie-2).
- 7. Learning Objectives At the end of the lesson, you should be able to do the following: 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, 𝐹1 and 𝐹2, is constant. Hyperbolas
- 9. 9 Parts and Properties of Hyperbolas Given arbitrary points 𝑃1 and 𝑃2 on the hyperbola, |𝑃1𝐹1 − 𝑃1𝐹2| = 𝑃2𝐹1 − 𝑃2𝐹2 = 2𝑎. What is the distance 𝑎?
- 10. 10 Parts and Properties of Hyperbolas 𝒄2 = 𝒂2 + 𝒃2
- 11. 11 How do you determine the value of 𝒂 given the values of 𝒃 and 𝒄? How do you determine the value of 𝒃 given the values of 𝒂 and 𝒄?
- 12. 12 Standard Equations of a Hyperbola Orientation of Principal Axis Equation Horizontal 𝒙 − 𝒉 𝟐 𝒂𝟐 − 𝒚 − 𝒌 𝟐 𝒃𝟐 = 𝟏 Vertical 𝒚 − 𝒌 𝟐 𝒂𝟐 − 𝒙 − 𝒉 𝟐 𝒃𝟐 = 𝟏
- 13. 13 Applications of Hyperbolas Hyperbolic Navigation Example: LORAN-C (long-range navigation)
- 14. 14 Applications of Hyperbolas Hyperbolic Navigation Two stations, 𝐴 and 𝐵, transmit radio signals. The radio signals reach the ship at different times.
- 15. 15 Applications of Hyperbolas Hyperbolic Navigation The location 𝑃 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 𝑃 is 𝑷𝑨 − 𝑷𝑩 = 𝟐𝒂.
- 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. Let’s Practice! 21 Point 𝑷 is on a hyperbola. The distance of 𝑷 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. Let’s Practice! 22 3 units Point 𝑷 is on a hyperbola. The distance of 𝑷 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. Try It! 23 23 Point 𝑷 is on a hyperbola. The difference between the distances of 𝑷 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. Let’s Practice! 24 What is the equation of a hyperbola whose center is at the origin, has a horizontal principal axis, the value of 𝒂 is 𝟒, and the point (𝟗, 𝟏𝟒) is on the hyperbola?
- 25. Let’s Practice! 25 𝒙𝟐 𝟏𝟔 − 𝒚𝟐 𝟒𝟖. 𝟐𝟓 = 𝟏 What is the equation of a hyperbola whose center is at the origin, has a horizontal principal axis, the value of 𝒂 is 𝟒, and the point (𝟗, 𝟏𝟒) is on the hyperbola?
- 26. Try It! 26 26 What is the equation of a hyperbola whose center is at the origin, has a vertical principal axis, the value of 𝒂 is 𝟖, and the point (𝟖, 𝟏𝟑) is on the hyperbola?
- 27. Let’s Practice! 27 Point 𝑷 is on a hyperbola with a vertical principal axis and whose center is at the origin. The distance of 𝑷 from the first focus 𝑭𝟏 is 12 units more than its distance from the second focus 𝑭𝟐. The distance between the two foci is 20 units. What is the equation of the hyperbola?
- 28. Let’s Practice! 28 𝒚𝟐 𝟑𝟔 − 𝒙𝟐 𝟔𝟒 = 𝟏 Point 𝑷 is on a hyperbola with a vertical principal axis and whose center is at the origin. The distance of 𝑷 from the first focus 𝑭𝟏 is 12 units more than its distance from the second focus 𝑭𝟐. The distance between the two foci is 20 units. What is the equation of the hyperbola?
- 29. Try It! 29 29 Point 𝑷 is on a hyperbola with a vertical principal axis and whose center is at the origin. The distance of 𝑷 from the first focus 𝑭𝟏 is 10 units more than its distance from the second focus 𝑭𝟐. The distance between the two foci is 26 units. What is the equation of the hyperbola?
- 30. Let’s Practice! 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. Let’s Practice! 31 𝒙𝟐 𝟔𝟐𝟓 − 𝒚𝟐 𝟗𝟑𝟎𝟗. 𝟎𝟗 = 𝟏 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. Try It! 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. Let’s Practice! 33 Two stations 𝑨 and 𝑩 are along a straight coast such that station 𝑨 is 100 miles west of station 𝑩. 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 𝑩 arrives at the ship 0.0002 of a second earlier than the radio signal sent from station 𝑨. Where is the ship? Set the midpoint of 𝑨 and 𝑩 as the origin.
- 34. Let’s Practice! 34 (𝟐𝟕. 𝟑𝟒, 𝟓𝟎) Two stations 𝑨 and 𝑩 are along a straight coast such that station 𝑨 is 100 miles west of station 𝑩. 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 𝑩 arrives at the ship 0.0002 of a second earlier than the radio signal sent from station 𝑨. Where is the ship? Set the midpoint of 𝑨 and 𝑩 as the origin.
- 35. Try It! 35 35 Two stations 𝑨 and 𝑩 transmit radio signals such that station A is 200 miles west of station 𝑩. Both stations sent radio signals with a speed of 𝟎. 𝟐 miles per microsecond to a plane traveling west. The signal from station 𝑩 reaches a plane 500 microseconds faster than the signal from station 𝑨. If the plane is 80 miles north of the line from stations 𝑨 to 𝑩, what is the location of the plane? Set the midpoint of 𝑨 and 𝑩 as the origin.
- 36. Check Your Understanding 36 Let 𝑷 be a point on a hyperbola with center at the origin and foci 𝑭𝟏 and 𝑭𝟐. Answer the following questions. Round off your answer to two decimal places. 1. If 𝑃𝐹1 − 𝑃𝐹2 = 18, what is 𝑎? 2. If 𝑃𝐹2 − 𝑃𝐹1 = 32 and 𝑏 = 12, what is 𝑐? 3. If 𝑃𝐹1 − 𝑃𝐹2 = 34 and 𝑏2 = 256, what is 𝑐? 4. If 𝑐 = 39 and 𝑎 = 36, what is 𝑏? 5. If 𝑐2 = 1 600 and 𝑎 = 32, what is 𝑏?
- 37. Check Your Understanding 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. Check Your Understanding 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. Check Your Understanding 39 Solve the following problems. 3. Two radio signaling stations 𝐴 and 𝐵 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 𝐵 arrives 0.0002 seconds before the signal from station 𝐴. What is the equation of the hyperbola where the ship is located? Set the midpoint of 𝐴 and 𝐵 as the origin.
- 40. Let’s Sum It Up! 40 ● A hyperbola is a set of points on a plane whose absolute difference between the distances from two fixed points 𝐹1 and 𝐹2 is constant. ● Given any point 𝑃 on a hyperbola with foci 𝐹1 and 𝐹2, 𝑃𝐹1 − 𝑃𝐹2 = 2𝑎.
- 41. Let’s Sum It Up! 41 ● The distance from the center to a vertex of the hyperbola is 𝒂, the distance from the center to an endpoint of the conjugate axis is 𝒃, and the focal distance is 𝒄. ● The relationship between the three distances is 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐.
- 42. Let’s Sum It Up! 42 ● The standard forms of equation of a hyperbola whose center is at the origin are 𝒙𝟐 𝒂𝟐 − 𝒚𝟐 𝒃𝟐 = 𝟏 if the principal axis is horizontal, and 𝒚𝟐 𝒂𝟐 − 𝒙𝟐 𝒃𝟐 = 𝟏 if the principal axis is vertical.
- 43. Let’s Sum It Up! 43 ● In the real world, hyperbolas are used for navigation and the structure of cooling towers.
- 44. Let’s Sum It Up! 44 ● To solve word problems on hyperbolas, follow these steps: 1. Determine the standard form of equation of the hyperbola. 2. Draw an illustration to be able to visualize the problem. 3. Solve for the unknown equation or values.
- 45. Key Formulas 45 Concept Formula Description Relationship between the values 𝒂, 𝒃, and 𝒄 𝑎2 + 𝑏2 = 𝑐2, where 𝑎 is the distance from the center to a vertex, 𝑏 is the distance from the center to an endpoint of the conjugate axis, and 𝑐 is the focal distance. Use this formula to find an unknown distance (e.g., focal distance) when the other values are known.
- 46. Key Formulas 46 Concept Formula Description Equation of a Hyperbola in Standard Form 𝑥 − ℎ 2 𝑎2 − 𝑦 − 𝑘 2 𝑏2 = 1, where (ℎ, 𝑘) is the center; 𝑎 is the distance from the center to a vertex, and 𝑏 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, 𝑎, and 𝑏 if the transverse axis is horizontal.
- 47. Key Formulas 47 Concept Formula Description Equation of a Hyperbola in Standard Form 𝑦 − 𝑘 2 𝑎2 − 𝑥 − ℎ 2 𝑏2 = 1, where (ℎ, 𝑘) is the center; 𝑎 is the distance from the center to a vertex, and 𝑏 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, 𝑎, and 𝑏 if the transverse axis is vertical.
- 48. Challenge Yourself 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. Photo Credits 49 ● Slides 2 and 3: Navigation boat Engineer Matusevich by Torin is licensed under CC BY-SA 3.0 via Wikimedia Commons. ● Slide 13: 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.
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