Chapter 6.1, Problem 57E

To determine

To find: the length of road from point A to E.

Answer to Problem 57E

  $5.31\text{ mi}$

Explanation of Solution

Given information:

The stretch of roadway is shown by the figure below, where the curves represent the arc of circles.

  

The values of length of line segment $\left(x\right)$ , central angle $\left(\theta \right)$ and radius $\left(r\right)$ of the circle given in the figure are as follows:

  $\begin{array}{ccc}\begin{array}{l}{\theta }_{AB}=84.5°\\ r_{{}_{AB}}=0.70\text{ mi}\end{array}& \begin{array}{l}{\theta }_{CD}=80°\\ r_{{}_{CD}}=0.67\text{ mi}\end{array}& \begin{array}{l}{x}_{BC}=1.46\text{ mi}\\ x_{DE}=1.8\text{ mi}\end{array}\end{array}$

Calculation:

Let the arc length of AB and CD be $s_{AB}$ and $s_{CD}$ respectively.

The arc length of AB is evaluated as follows:

  $\begin{array}{c}{s}_{AB}=r_{AB}{\theta }_{AB}\\ =\left(0.70\right)\left(84.5°\right)\left[\text{Put the value of }{\theta }_{AB}\text{ and }{r}_{AB}\right]\\ =\left(0.70\right)\left(84.5°\right)\left(\frac{\pi }{180°}\right)\left[\text{Convert }{\theta }_{AB}\text{ into radian measure}\right]\\ =1.03\text{ mi}\end{array}$

Thus the arc length of AB is $1.03\text{ mi}$ .

The arc length of CD is evaluated as follows:

  $\begin{array}{c}{s}_{CD}=r_{CD}{\theta }_{CD}\\ =\left(0.67\right)\left(80°\right)\left[\text{Put the value of }{\theta }_{CD}\text{ and }{r}_{CD}\right]\\ =\left(0.67\right)\left(80°\right)\left(\frac{\pi }{180°}\right)\left[\text{Convert }{\theta }_{CD}\text{ into radian measure}\right]\\ =0.94\text{ mi}\end{array}$

Thus the arc length of CD is $0.94\text{ mi}$ .

Now evaluate the length of road from point A to E as shown below.

  $\begin{array}{c}\text{Length of road}=s_{AB}+x_{BC}+s_{CD}+x_{DE}\\ =1.03+1.46+0.94+1.88\left[\text{Put the values}\right]\\ =5.31\text{ mi}\end{array}$

Thus the length of road from point A to E is $5.31\text{ mi}$ .