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A track is 2 meters wide and described by the equation (x^2)/(6400)+(y^2)/(900)=1....

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mpumpkin | Salutatorian

Posted June 7, 2013 at 11:51 AM via web

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A track is 2 meters wide and described by the equation (x^2)/(6400)+(y^2)/(900)=1.  Write the equation of the ellipse that forms the outer border.

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted June 7, 2013 at 12:19 PM (Answer #1)

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A track is 2m wide. The inside border is modeled by `x^2/6400 + y^2/900 = 1` . Find the equation of the ellipse for the outside border.

The general equation for an ellipse is `x^2/a^2+y^2/b^2=1` assuming the ellipse is centered at the origin. If a>b then the ellipse is longer along the horizontal axis. The longer axis is called the major axis (the shorter the minor axis). If a>b, then the major axis has length 2a and the semi-major axis (the distance from the center to the ellipse along the major axis) has length a.

Thus we have an ellipse with semi-major axis 80 and semi-minor axis 30. Since we have a track that is 2m wide, we add 2 to both the semi-major and semi-minor axes. (Or add 4 to the major and minor axes if you like.)

So the ellipse for the outer edge has semi-major axis length 82 and semi-minor axis length 32.

The equation will be `x^2/82^2 + y^2/32^2 =1`

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The equation is `x^2/6724 + y^2/1024=1`

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The graphs:

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