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Math Help - Ellipse

  1. #1
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    Ellipse

    A hallway 5.0 m wide has a ceiling whose cross section is a semi ellipse. The ceiling is 3.0 m high at the walls and 4.0 m high at the center. Find the height of the ceiling 1.0 m from each wall.

    How would I start this problem?
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  2. #2
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    Quote Originally Posted by OzzMan View Post
    A hallway 5.0 m wide has a ceiling whose cross section is a semi ellipse. The ceiling is 3.0 m high at the walls and 4.0 m high at the center. Find the height of the ceiling 1.0 m from each wall.

    How would I start this problem?
    From the measures of the hallway you know about the ellipse:

    semi-major axis : 2.5 m
    semi-minor axis : 1 m
    coordinates of the center: C(0, 3)

    Thus the equation of the ellipse is:

    \frac{x^2}{\left(\frac52\right)^2}+\frac{(y-3)^2}{1^2} = 1

    Because you only need the upper part of the ellipse you can solve for y to get the equation of a function:

    e(x) = 3+\sqrt{1-\frac{4x^2}{25}}

    The points A and B are 1 m from the wall: A(-1.5, e(1.5)) and B(1.5, e(1.5))
    Because the ellipse is symmetric about the y-axis it is only necessary to calculate e(1.5).

    I've got A(-1.5, 3.8), B(1.5, 3.8)
    Attached Thumbnails Attached Thumbnails Ellipse-ellipt_decke.gif  
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  3. #3
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    Oh I see. So my answers would be in the form of (x,y). Right?
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  4. #4
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    Quote Originally Posted by OzzMan View Post
    Oh I see. So my answers would be in the form of (x,y). Right?
    Not necessarily. You rae asked to calculate the height. It would be sufficient to give the answer as h = 3.8 m
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  5. #5
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    Quote Originally Posted by earboth View Post
    e(x) = 3+\sqrt{1-\frac{4x^2}{25}}
    Shouldn't it be e(y) = 3+\sqrt{1-\frac{4x^2}{25}}

    And also how come you have to put an e in there?
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  6. #6
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    Quote Originally Posted by OzzMan View Post
    Shouldn't it be e(y) = 3+\sqrt{1-\frac{4x^2}{25}}

    And also how come you have to put an e in there?
    I transformed the equation of the ellipse into the equation of a function whose graph is the upper part of the ellipse.

    Therefore:

    y = e(x) = 3+\sqrt{1-\frac{4x^2}{25}}

    which will yield the height above the x-axis.
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