1013 10.2 Ellipses 56. Height and Width of an Overpass An arch has the shape of half an ellipse. The equation of the ellipse is 100x2 + 324y2 = 32,400, where x and y are in meters. y x NOT TO SCALE (a) How high is the center of the arch? (b) How wide is the arch across the bottom? 57. Orbit of Halley’s Comet The famous Halley’s comet last passed by Earth in February 1986 and will next return in 2062. It has an elliptical orbit of eccentricity 0.9673 with the sun at one focus. The greatest distance of the comet from the sun is 3281 million mi. Find the least distance between Halley’s comet and the sun to the nearest million miles. (Data from The World Almanac and Book of Facts.) 58. (Modeling) Orbit of a Satellite The coordinates in miles for the orbit of the artificial satellite Explorer VII can be modeled by the equation x2 a2 + y2 b2 = 1, where a = 4465 and b = 4462. Earth’s center is located at one focus of the elliptical orbit. (Data from Loh, W., Dynamics and Thermodynamics of Planetary Entry, Prentice-Hall; Thomson, W., Introduction to Space Dynamics, John Wiley and Sons.) (a) Graph both the orbit of Explorer VII and the Earth’s surface on the same coordinate axes if the average radius of Earth is 3960 mi. Use the window 3-6750, 67504 by 3-4500, 45004. (b) Find the maximum and minimum heights of the satellite above Earth’s surface to the nearest mile. 59. (Modeling) Orbits of Satellites Neptune and Pluto both have elliptical orbits with the sun at one focus. Neptune’s orbit has a = 30.1 astronomical units (AU) with an eccentricity of e = 0.009, whereas Pluto’s orbit has a = 39.4 and e = 0.249. (Data from Zeilik, M., S. Gregory, and E. Smith, Introductory Astronomy and Astrophysics, Fourth Edition, Saunders College Publishers.) (a) Position the sun at the origin and determine equations that model each orbit. (b) Graph both equations on the same coordinate axes. Use the window 3-60, 604 by 3-40, 404. 60. (Modeling) The Roman Colosseum (a) The Roman Colosseum is an ellipse with major axis 620 ft and minor axis 513 ft. Find the distance between the foci of this ellipse to the nearest foot. a b (b) A formula for the approximate perimeter of an ellipse is P ≈2pBa2 + b2 2 , where a and b are the lengths shown in the figure. Use this formula to find the perimeter of the Roman Colosseum to the nearest foot.
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