Arailroad tunnel is shaped like a semi ellipse, as shown below. 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.

Remark
The general equation for an ellipse is
 

At the center, the the point is 0,27

Solving for the 

or b^2= 27^2 * 1
Taking the square root of both sides 
b = 27

Now we need to use one more piece of information
The second point is (24,9) what that means is at the point x = 24, y = 9

Solving for that 


576/a^2 + 81/729 = 1    Subtract 81/729 from both sides
576/a^2 = 1 - 81/729 = 576/a^2 = (729 - 81)/729 = 648 /729

576/a^2 = 648/729     Cross multiply

648 a^2 = 729 * 576   Divide by 648
a^2 = 648            

Full equation for the ellipse and answer is