Highway design. in order to build a highway, it is necessary to fill a section of a valley where the grades (slopes) of the sides are 9% and 6% (see figure). the top of the filled region will have the shape of a parabolic arc that is tangent to the two slopes at the points a and b. the horizontal distances from a to the -axis and from b to the -axis are both 500 feet. what will be the lowest point on the completed highway?

Since the diagram is not posted, we will assume point A is to the left of the y-axis and has a slope of -9%.  Point B is to the right of the y-axis and has a slope of +6%.

Further, we will assume the equation of the parabola is 
y=ax^2+bx+c
and that the asker is familiar with elementary differential calculus.

We find the slope of the parabola by differentiation.
y'(x)=2ax+b
From the given information, we substitute at point A,
Slope at A = y'(-500)=2a(-500)+b=-0.09(1)
Slope at B = y'(+500)=2a(500)+b=+0.06(2)

Solve (1) and (2) for a and b,
(2)-(1) : 2000a=0.15 => a=0.000075
  Note: a>0 means that the parabola has a relative minimum.

substitute a in (1) => b= -0.015   

Finally, to find the lowest point (relative minimum)
we set y'(x)=0, or
2ax+b=0,
substitute a=0.000075, and b=-0.015 to get
2(0.000075)x-0.015=0
Solve for x gives
x=0.015/(0.000150)=100 
The lowest point is at x=100, or between A and B, 600 ft from A.