SOLUTION: An arch of a bridge over a highway is semi-elliptical in shape and 50 feet across. The highest point of the arch is 15 feet above the highway. What is the maximum height of a vehic

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: An arch of a bridge over a highway is semi-elliptical in shape and 50 feet across. The highest point of the arch is 15 feet above the highway. What is the maximum height of a vehic      Log On


   

Question 1171519: An arch of a bridge over a highway is semi-elliptical in shape and 50 feet across. The highest point of the arch is 15 feet above the highway. What is the maximum height of a vehicle 10 feet wide that can fit under arch ?
Answer by ikleyn(52781) About Me  (Show Source):
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An arch of a bridge over a highway is semi-elliptical in shape and 50 feet across.
The highest point of the arch is 15 feet above the highway.
What is the maximum height of a vehicle 10 feet wide that can fit under arch ?
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The major semi-axis is  a = 50/2 = 25 ft, horizontally.


The minor semi-axis is b = 15 ft, vertically.


The equation of the ellipse is

    x%5E2%2F25%5E2 + y%5E2%2F15%5E2 = 1.


From this equation

    y = 15%2Asqrt%281+-+%28x%2F25%29%5E2%29.


At x = 10/2 = 5 ft,  the y-coordinate is

    y = 15%2Asqrt%281+-+%285%2F25%29%5E2%29 = 15%2Asqrt%281+-+%281%2F5%29%5E2%29 = 15%2Asqrt%281-0.2%5E2%29 = 15%2Asqrt%280.96%29 = 14.697 ft.


ANSWER.  The maximum height to fit is 14.6 ft  (reasonably and providently rounded down).

Solved.