Chapter 9.1, Problem 98E

(a)

To determine

To find:The equation of the parabola with its vertex at origin that models the road surface.

Answer to Problem 98E

The equation of the parabola is $x^2=-640y$ .

Explanation of Solution

Given information:

Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is $32$ feet wide is $0.4$ feet higher in the center than it is on the sides.

  

Calculation:

As clear from the figure given above the parabola is vertically downwards with vertex at origin and also it passes through the point $\left(16,-0.4\right)$ .

The general equation of the parabola is $x^2=4py$ .

Now substitute $16$ for $x$ and $-0.4$ for $y$ in the equation.

  $\begin{array}{c}{x}^2=4py\\ {16}^2=4p\left(-0.4\right)\\ p=-\frac{256}{0.16}\\ p=-160\end{array}$

Now substitute $-160$ for $p$ in the equation of the parabola.

  $\begin{array}{c}{x}^2=4py\\ x^2=4\left(-160\right)y\\ x^2=-640y\end{array}$

Therefore, the equation of the parabola is $x^2=-640y$ .

(b)

To determine

To calculate:The distance of the surface from the center of the road at the time of surface being $0.1$ foot lower than in the middle.

Answer to Problem 98E

The distance of the surface from the center of the road at the time of surface being $0.1$ foot lower than in the middle is $13.86$ foot.

Explanation of Solution

Given information:

Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is $32$ feet wide is $0.4$ feet higher in the center than it is on the sides.

  

Calculation:

As calculated in part(a), the equation of the parabola is $x^2=-640y$ .

Now the height of the surface is $0.1$ foot lower than in the middle.

So,

  $\begin{array}{c}y=0.4-0.1\\ =0.3\end{array}$ .

Now substitute $-0.3$ for $y$ in the equation of the parabola.

  $\begin{array}{c}{x}^2=-640y\\ x^2=-640\left(-0.3\right)\\ x^2=192\\ x=\pm 13.86\end{array}$

Therefore, the distance of the surface from the center of the road at the time of surface being $0.1$ foot lower than in the middle is $13.86$ foot.