# 27 the orbit of halleys comet last seen in 1986 and

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27. The orbit of Halley’s comet, last seen in 1986 and due to return in 2062, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is 36.18 AU. [An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles.] Find a polar equa- tion for the orbit of Halley’s comet. What is the maximum distance from the comet to the sun? 28. The Hale-Bopp comet, discovered in 1995, has an elliptical orbit with eccentricity 0.9951 and the length of the major axis is 356.5 AU. Find a polar equation for the orbit of this comet. How close to the sun does it come? 29. The planet Mercury travels in an elliptical orbit with eccen- tricity . Its minimum distance from the sun is km. Find its maximum distance from the sun. 30. The distance from the planet Pluto to the sun is km at perihelion and km at aphelion. Find the eccentricity of Pluto’s orbit. 31. Using the data from Exercise 29, find the distance traveled by the planet Mercury during one complete orbit around the sun. (If your calculator or computer algebra system evaluates defi- nite integrals, use it. Otherwise, use Simpson’s Rule.) e y d r ed 1 e sin e y d r ed 1 e sin r c 1 cos r d 1 cos 0.093 2.28 10 8 km 0.048 1.56 10 9 km 0.206 4.6 10 7 4.43 10 9 7.37 10 9 10.6 Exercises ; Graphing calculator or computer required 1. Homework Hints available at stewartcalculus.com © Dean Ketelsen
CHAPTER 10 REVIEW 709 10 Review 1. (a) What is a parametric curve? (b) How do you sketch a parametric curve? 2. (a) How do you find the slope of a tangent to a parametric curve? (b) How do you find the area under a parametric curve? 3. Write an expression for each of the following: (a) The length of a parametric curve (b) The area of the surface obtained by rotating a parametric curve about the 4. (a) Use a diagram to explain the meaning of the polar coordi- nates of a point. (b) Write equations that express the Cartesian coordinates of a point in terms of the polar coordinates. (c) What equations would you use to find the polar coordinates of a point if you knew the Cartesian coordinates? 5. (a) How do you find the slope of a tangent line to a polar curve? (b) How do you find the area of a region bounded by a polar curve? (c) How do you find the length of a polar curve? x -axis r , x , y 6. (a) Give a geometric definition of a parabola. (b) Write an equation of a parabola with focus and direc- trix . What if the focus is and the directrix is ? 7. (a) Give a definition of an ellipse in terms of foci. (b) Write an equation for the ellipse with foci and vertices . 8. (a) Give a definition of a hyperbola in terms of foci. (b) Write an equation for the hyperbola with foci and vertices . (c) Write equations for the asymptotes of the hyperbola in part (b). 9. (a) What is the eccentricity of a conic section? (b) What can you say about the eccentricity if the conic section is an ellipse? A hyperbola? A parabola?