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Math Help - [SOLVED] An ugly PP calc word problem

  1. #1
    Junior Member xterminal01's Avatar
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    [SOLVED] An ugly PP calc word problem

    I found this on a power point on the net and trying to solve it.
    Its a conic section from the ellipse chapter.



    • The mean distance of Mars from the Sun is 142 million miles.
      • Perihelion = 128.5 million miles
      • 1. What is Aphelion = ??
      • 2. Equation for Mars orbit?
      • Round to nearest tenth if necessary.
    • Hint: Orbit of a planet about the sun is an Ellipse, with the Sun at one focus. Aphelion of a planet is its greatest distance from the Sun and the perihelion is its shortest distance. Mean distance of a planet from the sun is the length of the semi major axis of the elliptical orbit.

    1. Aphelion
    2. Equation
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  2. #2
    MHF Contributor
    Joined
    Apr 2005
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    The equation of an ellipse, whose center is at (0,0):
    [(x^2)/(a^2)] +[(y^2)/(b^2)] = 1 ------------(i)
    Also,
    b^2 = a^2 -c^2 -----------------------------(ii)
    where
    a = semi-major axis ----mean distance in this Problem.
    b = semi-minor axis
    c = focal distance, from the center
    (x,y) is any point on the ellipse.

    So,
    perihelion +aphelion = 2a = 2(mean distance)
    128.5 +aphelion = 2(142)
    aphelion = 284 -128.5 = 155.5 million miles -------answer.

    -----------------------------
    Equation of Mars' orbit:

    c = "a" minus perihelion = mean distance minus perihelion ----(1)
    c = 142 -128.5 = 13.5 million miles

    Hence,
    b^2 = (142)^2 -(13.5)^2
    b^2 = 19,981.75 --------------------------(***)

    Therefore, the equation of Mars' orbit is
    [(x^2)/(142^2)] +[(y^2)/(19,981.75)] = 1
    [(x^2)/(20,164)] +[(y^2)/(19,981.75)] = 1 -------------answer.
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