Question 1171519
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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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<pre>
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^2/25^2}}} + {{{y^2/15^2}}} = 1.


From this equation

    y = {{{15*sqrt(1 - (x/25)^2)}}}.


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

    y = {{{15*sqrt(1 - (5/25)^2)}}} = {{{15*sqrt(1 - (1/5)^2)}}} = {{{15*sqrt(1-0.2^2)}}} = {{{15*sqrt(0.96)}}} = 14.697 ft.


<U>ANSWER</U>.  The maximum height to fit is 14.6 ft  (reasonably and providently rounded down).
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Solved.