17.2k views
4 votes
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.

1 Answer

2 votes
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
The height of the tunnel at the center is 50ft, and the vertical clearance must be-example-1
User DerSteve
by
8.6k points