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