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?
Accepted Solution
A:
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.