Chapter 8.1, Problem 67AYU

Constructing a Highway A highway whose primary directions are north-south is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of highway needed to go around the bay?

To determine

To find: What is the length of highway needed to go around the bay?

Answer to Problem 67AYU

The length of the highway is $3.83$ miles.

Explanation of Solution

Given:

A highway whose primary directions are north–south is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The illustration shows the path that they decide on and the measurements taken.

Formula used:

$\text{cos}\theta =\text{ adj}/\text{hyp}$

$\text{sin}\theta =\text{ opp}/\text{hyp}$

Calculation:

From the given figure we construct the following figure.

$\left(180-130=50°\text{ and }180°-140°=40°\right)$

Let $x$, $y$ and $z$ be the distance as shown in the figure.

To find $x+y+z$.

From the figure

$\sin 40°=\frac{1}{x}$

$x=1/\sin 40°$

$x=1.5557$ miles.

Similarly, $z=\frac{1}{\sin 50°}=1.3054$.

To find $y$, we need find $a$ and $b$.

$\cos 50=\frac{b}{z}=b=z\text{cos}50°=1.3054*\text{ cos}50°\approx 0.8391$

$\text{cos}40°=a/x$

$a=1.5557*\text{ cos}40°=1.918$ miles.

$y=3-a-b=3-1.918-0.8391=0.9691$

$y=0.9691$ miles.

$x+y+z=1.5557+0.9691+1.3054\approx 3.83$ miles.

The length of the highway is $3.83$ miles.