734 CHAPTER 10 Analytic Geometry In Problems 13–24, analyze each equation and graph it. 13. r 1 1 cosθ = + 14. r 3 1 sinθ = − 15. r 8 4 3sinθ = + 16. r 10 5 4 cosθ = + 17. r 9 3 6cosθ = − 18. r 12 4 8 sinθ = + 19. r 8 2 sinθ = − 20. r 8 2 4cosθ = + 21. r 3 2 sin 6 θ ( ) − = 22. r 2 cos 2 θ ( ) − = 23. r 6 sec 2 sec 1 θ θ = − 24. r 3csc csc 1 θ θ = − In Problems 25–36, convert each polar equation to a rectangular equation. 25. r 1 1 cosθ = + 26. r 3 1 sinθ = − 27. r 8 4 3sinθ = + 28. r 10 5 4cosθ = + 29. r 9 3 6cosθ = − 30. r 12 4 8 sinθ = + 31. r 8 2 sinθ = − 32. r 8 2 4cosθ = + 33. r 3 2 sin 6 θ ( ) − = 34. r 2 cos 2 θ ( ) − = 35. r 6 sec 2 sec 1 θ θ = − 36. r 3csc csc 1 θ θ = − In Problems 37–42, find a polar equation for each conic. For each, a focus is at the pole. 37. e 1; = directrix is parallel to the polar axis, 1 unit above the pole. 38. e 1; = directrix is parallel to the polar axis, 2 units below the pole. 39. e 4 5 ; = directrix is perpendicular to the polar axis, 3 units to the left of the pole. 40. e 2 3 ; = directrix is parallel to the polar axis, 3 units above the pole. 41. e 6; = directrix is parallel to the polar axis, 2 units below the pole. 42. e 5; = directrix is perpendicular to the polar axis, 5 units to the right of the pole. Applications and Extensions 43. Derive r ep e 1 cosθ = + from Table 5. 44. Derive r ep e 1 sinθ = + from Table 5. 45. Derive r ep e 1 sinθ = − from Table 5. 46. Orbit of Mercury The planet Mercury travels around the Sun in an elliptical orbit given approximately by r 3.442 10 1 0.206cos 7 θ = × − where r is measured in miles and the Sun is at the pole. Find the distance from Mercury to the Sun at aphelion (greatest distance from the Sun) and at perihelion (shortest distance from the Sun). See the figure. Use the aphelion and perihelion to graph the orbit of Mercury using a graphing utility. Perihelion Mercury Aphelion Sun 47. Halley’s Comet Halley’s comet travels around the Sun in an elliptical orbit given approximately by r 1.155 1 0.967cosθ = − where the Sun is at the pole and r is measured in AU (astronomical units). Find the distance from Halley’s comet to the Sun at aphelion and at perihelion. Use the aphelion and perihelion to graph the orbit of Halley’s comet using a graphing utility. 48. Challenge Problem Water Leak A tank is punctured on its side, and water begins to stream out in a parabolic path. If the path of the water is given by r 0.8 1 sinθ = + and the water hits the ground 4 inches away from the base of the tank, what is the height of the puncture from the base of the tank? Assume the focus is at the pole. Puncture height 4 in.
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