The height of the tunnel at the center is 50ft, and the vertical clearance must be 12ft at a point 30ft from the center. find an equation for the ellipse.

Please refer to the illustration for this problem shown in the picture. 

Imagine that the tunnel is the upper half of an ellipse drawn on a Cartesian plane. The general equation for this ellipse is:

x²/a² + y²/b² = 1, where

a is the semi-major axis
b is the semi-minor axis

In this ellipse, the major axis and the minor axes are the x and y, respectively. We know a value of 50 ft in the y-axis. The distance from the center to the highest point of the ellipse is the semi-minor axis. Therefore, b = 50 ft. Now, we substitute point (30,12) to the general from to determine a.

30²/a² + 12²/50² = 1
a² = 955

So, the equation of the ellipse is

x²/955 + y²/50 = 1