Question

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

Answer 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