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Math Help - Another Ellipse Problem

  1. #1
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    Unhappy Another Ellipse Problem

    The Earth's orbit is an ellipse with the sun at one focus. The length of the major axis is 186,000,000 miles and the eccentricity 0.0167. Find the distances from the ends of the major axis to the sun. These are the greatest and least distances from the earth to the sun.

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  2. #2
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    The Earth's orbit is an ellipse with the sun at one focus. The length of the major axis is 186,000,000 miles and the eccentricity 0.0167. Find the distances from the ends of the major axis to the sun. These are the greatest and least distances from the earth to the sun.
    Hello,

    the foci have the distance e (called linear eccentricity (literally translated from German)) from the centre of the ellipse. The proportion \epsilon=\frac{e}{a} is called numerical eccentricity (translated ... as you now know) and that's the value you are given here.

    Thus you can calculate the distance e:

    0.0167 = \frac{e}{93,000,000} \Longleftrightarrow e \approx 1,553,100 \text{ miles}

    If you subtract this value from the half axis you'll get the shortest distance (it is called perihel) p = 91,446,900 miles

    If you add this value to the half axis you'll get the greatest distance (it is called aphel) ap = 94,553,100 miles

    EB

    PS: Make a sketch of an ellipse and mark the lines a, b, e, p and ap.
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  3. #3
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    The Earth's orbit is an ellipse with the sun at one focus. The length of the major axis is 186,000,000 miles and the eccentricity 0.0167. Find the distances from the ends of the major axis to the sun. These are the greatest and least distances from the earth to the sun.

    Umm, eccentricity in a conic section.

    Eccentricity, e, is the ratio of the distances of a point on the curve from the focus and from the directrix. If f = distance from focus, and d = distance from directrix, then e = f/d.

    Also, e = c/a -----------(ii)
    where
    c = distance of focus from the center of ellipse
    a = distance of end (vertex) of ellipse from the center along the major axis, or it is half of the major axis.

    Given:
    2a = 186M miles.
    e = 0.0167 ----------very small. So the ellipse is almost like a circle.

    Using (ii),
    0.0167 = c/(186/2)
    c = (0.0167)(93) = 1.5531M mi.

    The focus is along the major axis, so,
    f = 93 -1.5531 = 91.4469M mi.
    Meaning, the nearest vertex of the ellipse to sun is 91.4469 million miles.
    The other vertex is 186M -91.4469M = 94.5531M miles from the sun.

    Therefore, the least distance of the Earth to the sun is about 91,446,900 miles, and the Earth's greatest distance from the sun is about 94,553,100 miles. -----------answer.
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