Chapter 1.9, Problem 72PS

Solution

To Find: The distance $\left(x\right)$ of the arch above the road.

The distance of the arch above the road is $55.1<x<447.3\left(\text{meters}\right)$ .

Given information:

The given scenario is modelled by $y=-0.00211x^2+1.06x$

Distance of the bridge which the arc is located $x=52$

  $x$ is the distance in meters.

  $y$ is the height in meters.

Calculation:

The objective is to find the distance of the arch above the road.

Given the model:

  $y=-0.00211x^2+1.06x$

We need to consider the arc above the road which is $52$. So, the inequality becomes:

  $-0.00211x^2+1.06x>52$

Rewrite the inequality as:

  $0>0.00211x^2-1.06x+52$

Find the zeros with the quadratic formula.

  $\begin{array}{l}0=0.00211x^2-1.06x+52\\ x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-\left(-1.06\right)\pm \sqrt{{\left(-1.06\right)}^2-4\left(0.00211\right)\left(52\right)}}{2\left(0.00211\right)}\\ x=\frac{1.06\pm \sqrt{0.68472}}{0.00422}\\ x\approx 55.1,447.3\end{array}$

Since everything was moved to the other side to get one side to $0$ , $y=0.00211x^2-1.06x+52$ is an upward opening parabola. The part below the x-axis is the part between the zeros.

  $0.00211x^2-1.06x+52<0$

There is an arch above the road between the zeros (excluding them).

  $55.1<x<447.3$