Chapter 8, Problem 39P

A proposed highway traverses a hill-top bounded by uphill and downhill grades of 15% and -10%, respectively. These grades pass through benchmarks A and B located as shown in Fig. P8.39.

With the origin of the coordinate axes (x, y) set at benchmark A, the engineer has defined the hilltop segment of the

highway by a parabolic arc $y\left(x\right)=a{x}^2+bx$ which is tangent to the uphill grade at the origin.

(a) Find the slope of the line for the up-hill grade and the value of b for the parabolic arc.

(b) Find the equation of the line

$\hat{y}=cx+d$ for the downhill grade.

(c) Given that at the downhill point of tangency $\left(\bar{x}\right)$, both the elevation and the slope of the parabolic arc are equal to their respective value of the downhill line, for example,

$y\left(\bar{x}\right)=\hat{y}\left(\bar{x}\right)$ $\frac{dy}{dx}{|}_{x=\bar{x}}=\frac{d\hat{y}}{d{x}_{x=\bar{x}}}$

determine the point of tangency $\left(\bar{x},\bar{y}\right)$ of the parabolic arc with the downhill grade. Also, compute, its elevation.

(d) Find the equation of the parabolic arc.