Trig

A railroad tunnel is shaped like a semiellipse as shown below.

A semiellipse is shown on the coordinate plane with vertices on the x axis and one point of intersection with the positive y axis.

The height of the tunnel at the center is 27 ft and the vertical clearance must be 9 ft at a point 24 ft from the center. Find an equation for the ellipse

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  1. clearly,

    x^2/a^2 + y^2/27^2 = 1
    at x=24, y=9, so

    24^2/a^2 + 9^2/27^2 = 1
    24^2/a^2 + 1/9 = 1
    24^2/a^2 = 8/9
    a^2 = 24^2 * 9/8 = 648

    x^2/648 + y^2/729 = 1

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