Chapter 16, Problem 8RP

To determine

To find: the four significant digits, the distance from the point to the graph.

Answer to Problem 8RP

  $d=6.324$ .

Explanation of Solution

Given:

The graph of the function is $x-3y=7$ .

The point is $\left(-2,5\right)$ .

Concept used:

The equation of straight line having one point is:

  $y=mx+c$ .

  $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$ .

Calculation:

The graph of the function is $x-3y=7$ can be written as:

  $y=\frac{1}{3}x-\frac{7}{3}.$

The equation of straight line having one point is:

  $y=mx+c$ .

slope $=\frac{1}{3}$ .

The slope of the line that will give the shortest distance from the point to the given line, find the coordinate of the point into the formula for a line to get the equation as:

  $\begin{array}{l}\left(5\right)=-3\left(-2\right)+c.\\ c=-1.\\ y=-3x-1.\end{array}$

So, the new point will be $\left(0,-1\right)$ .

The given point is $\left(-2,5\right)$ .

The distance formula:

  $\begin{array}{l}d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}.\\ d=\sqrt{{\left(0-\left(-2\right)\right)}^2+{\left(\left(-1\right)-5\right)}^2}.\\ d=\sqrt{{\left(2\right)}^2+{\left(-6\right)}^2}.\\ d=\sqrt{40}=\mathrm{6.324.}\end{array}$

Hence, the distance from the point will be $d=6.324$ .