Question

The height of the tunnel at the center is 54 ft, and the vertical clearance must be 18 ft at a point 8 ft from the center. Find an equation for the ellipse.

Answer 1

the standard form for a horizontal ellipse is

X^2/a^2 + y^2/b^2 = 1

Substitute 54 for b and and use (8,18) as the

point to find a

x=8

y=18

8^2/a^2 + 18^2/54^2 =1

64/a^2 +  324/2916 = 1

324/2916 reduces to 1/9

64/a^2 + 1/9 = 1

64/a^2= 1-1/9

64/a^2 = 8/9

64*9/8 = a^2

576/8 = 72

A^2 = 72

A = square root(72) = 8.485

 So formula

would be x^2/72 + y^2/2916 =1