Pre Calculus Ellipses Question

Question

A certain bridge arch is in the shape of the top half pf an ellipse 116 feet wide and 31.4 feet high. At what horizontal distance from the center of the arch is the height equal to 12.2 feet? Give your answer accurate to 2 decimal places.

Answer 1

Draw a picture of the situation. I think this is what you describe:

Let the origin be at the center. Then to each side (x-direction) it is 118/2 or 58 ft so 58² is what goes under the x². In the y-direction, it is 31,4 to the top so 31.4² goes under the y² So the equation of the ellipse that models this arch is:

x²/58² + y²/31.4² = 1

To find the x value where the height (y-coordinate) is 12.2, we just plug in 12.2 for y and solve for x.

x²/58² + (12.2)²/31.4² = 1

x²/3364 + 148.84/985.86 = 1

x²/3364 + 0.15096 = 1

x²/3364 = 1 - 0.15096

x²/3364 = 0.84904

x² = (0.84904)(3364)

x² = 2856.172

x = √2856.172 = 53.44 ft to either side of the center