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Conics: Ellipses: Word Problems (page 4 of 4) Sections: Introduction, Finding information from the equation, Finding the equation from information, Word Problems
Since the ceiling is half of an ellipse (the top half, specifically), and since the foci will be on a line between the tops of the "straight" parts of the side walls, the foci will be five feet above the floor, which sounds about right for people talking and listening: five feet high is close to facehigh on most adults. I'll center my ellipse above the origin, so (h, k) = (0, 5). The foci are thirty feet apart, so they're 15 units to either side of the center. In particular, c = 15. Since the elliptical part of the room's crosssection is twenty feet high above the center, and since this "shorter" direction is the semiminor axis, then b = 20. The equation b^{2} = a^{2} – c^{2} gives me 400 = a^{2} – 225, so a^{2} = 625. Then the equation for the elliptical ceiling is: I need to find the height of the ceiling above the foci. I prefer positive numbers, so I'll look at the focus to the right of the center. The height (from the ellipse's central line through its foci, up to the ceiling) will be the yvalue of the ellipse when x = 15:
(Since I'm looking for the height above,
not the depth below, I ignored the negative solution to the quadratic
equation.) Copyright
© Elizabeth Stapel 20102011 All Rights Reserved The ceiling is 21 feet above the floor.
The lowest altitude will be at the vertex closer to the Earth; the highest altitude will be at the other vertex. Since I need to measure these altitudes from the focus, I need to find the value of c. b^{2} = a^{2} – c^{2}c^{2} = a^{2} – b^{2} = 4420^{2} – 4416^{2} = 35,344 Then c = 188. If I set the center of my ellipse at the origin and make this a widerthantall ellipse, then I can put the Earth's center at the point (188, 0). (This means, by the way, that there isn't much difference between the circumference of the Earth and the path of the satellite. The center of the elliptical orbit is actually inside the Earth, and the ellipse, having an eccentricity of e = 188 / 4420, or about 0.04, is pretty close to being a circle.) The vertex closer to the end of the ellipse containing the Earth's center will be at 4420 units from the ellipse's center, or 4420 – 188 = 4232 units from the center of the Earth. Since the Earth's radius is 3960 units, then the altitude is 4232 – 3960 = 272. The other vertex is 4420 + 188 = 4608 units from the Earth's center, giving me an altitude of 4608 – 3960 = 648 units. The minimum altitude
is 272
miles above the Earth; << Previous Top  1  2  3  4  Return to Index



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