Chapter 1.3, Problem 31E

To determine

To find: The shortest distance you must travel to reach the river.

Answer to Problem 31E

The shortest distance is 2.21 miles.

Explanation of Solution

Given Information: The diagram which shows the position of river that cuts through the woods. You are currently 2 miles east and 1 mile north of your starting point, the origin.

The shortest path of river will be perpendicular to the river Slope of river $=\frac{-1}{3}$

We are at position (2,1)

Equation of line

  $\begin{array}{l}y-y_1=m(x-x_1)\\ y-0=\frac{-1}{3}(x-2)\\ y-1=\frac{-x}{3}+\frac{2}{3}\\ y=\frac{-x}{3}+\frac{2}{3}+1\\ y=\frac{-x}{3}+\frac{5}{3}\end{array}$

Now slope the equation

  $\begin{array}{l}\frac{-x}{3}+\frac{5}{3}=3x+2\\ \frac{-x}{3}-\frac{3x}{1}=2-\frac{5}{3}\\ \frac{-x-9x}{3}=\frac{6-5}{3}\\ \frac{-10x}{3}=\frac{1}{3}\\ -10x=1\\ x=\frac{-1}{10}\end{array}$

  $\begin{array}{l}y=\frac{-1}{3}(\frac{-1}{10})+\frac{5}{3}\\ y=\frac{1}{30}+\frac{5}{3}\\ y=\frac{1+50}{30}=\frac{51}{30}\\ y=\frac{17}{10}\end{array}$

Shortest distance

  $\begin{array}{l}=\sqrt{{(\frac{-1}{10}-2)}^2+{(\frac{17}{10}-1)}^2}\\ =\sqrt{{(\frac{-1-20}{10})}^2+{(\frac{17-10}{10})}^2}\\ =\sqrt{{(\frac{-21}{10})}^2+{(\frac{7}{10})}^2}\\ =\sqrt{\frac{441}{100}+\frac{49}{100}}\\ =\sqrt{\frac{49}{10}}\\ =2.21\end{array}$

Thus,The shortest distance is 2.21 miles.