SOLUTION: A parabolic arch 27 feet high spans a parkway. How wide is the arch if the center section of the parkway, a section that is 50 feet wide, has a minimum clearance of 15 feet?

Let the parabola have vertex (0,27), and let the 50-foot parkway extend
from (-25,0) to (25,0). To have minimum clearance at the edges of 15 feet,
the parabola must pass through points (-25,15) and (25,15).



The equation of a parabola has several forms,  I'll use this one:



where (h,k) is the vertex (0,27)





Since it passes through (25,15), we substitute







So the equation of the parabola is





To find the width of the entire arch at the
bottom, we must find the x-intercepts. So we 
substitute 0 for y:






So the width of the arch at the bottom goes from
(-37.5,0) to {37.5,0). 

And so the width of the arch at the bottom is 

(37.5)(2) = 75 feet.

Edwin