Chapter 10.5, Problem 19E

To determine

To find how far is the gazebo from the nature trail.

Answer to Problem 19E

  $95\text{ unit}$

Explanation of Solution

Given:

An equation of the line representing the nature trail is $y=\frac{1}{3}x-4$ . Each unit in the coordinate plane corresponds to 10 feet

Calculation:

It is given that gazebo is at $(-6,4)$ and the line is $y=\frac{1}{3}x-4$ .

  

The slope of the line perpendicular to the line $y=\frac{1}{3}x-4$ is $m=-3$ .

Therefore the equation of line passing through $(-6,4)$ is −

  $\begin{array}{l}y=mx+c\\ 4=-3(-6)+b\\ b=-14\end{array}$

So the equation of line perpendicular to $y=\frac{1}{3}x-4$ is $y=-3x-14$

Solving the above two equations −

  $\begin{array}{l}y=\frac{1}{3}x-4\\ y=-3x-14\end{array}$

The solution obtained i.e. point of intersection is $(-3,-5)$ .

Now using the distance formula to calculate the distance between $(-3,-5)$ and $(-6,4)$ -

Distance $\begin{array}{l}=\sqrt{{(-6+3)}^2+{(4+5)}^2}\\ =\sqrt{{(-3)}^2+{(9)}^2}\\ =\sqrt{90}\\ =9.5\text{ unit}\end{array}$

Since ach unit in the coordinate plane corresponds to 10 feet, therefore the distance is $95\text{ unit}$ .