Question

A railroad tunnel is shaped like semi-ellipse. The height of the tunnel at the center is 69 ft and the vertical clearance must be 23 ft at a point 16ft from the center. Find an equation for the ellipse.

Answer 1

b = 69, a is unknown, and the point (16,23) is on it
let the equation be
x^2/a^2 + y^2/69^2 = 1
16^2/a^2 + 23^2/69^2 = 1
256/a^2 = 1 - 529/4761
256/a^2 = 8/9
8a^2 = 2304
a^2 = 288

x^2/288 + y^2/4761 = 1