Chapter 8.1, Problem 53PS

Solution

To find: the coordinates of $M.$

The coordinates of $M$ are $\left(-1.5,8\right)$ .

Given:

The diagram of a trail system is given below:

  

Formula Given:

Midpoint formula for calculating the mid-point of a line segment whose end points are $(x_1,y_1)$ and $(x_2,y_2)$ , is given as:

  $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Calculation:

  $M$is the mid-point for the line segment with two endpoints $S$ and $P$ . Now, calculating the mid-point using midpoint formula as:

  $\begin{array}{l}M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ M=\left(\frac{-6+3}{2},\frac{5+11}{2}\right)\\ M=\left(\frac{-3}{2},\frac{16}{2}\right)\\ M=\left(-1.5,8\right)\end{array}$

Hence, the coordinates of $M$ are $\left(-1.5,8\right).$

To find: the total distance travelled from $V\text{ to }M.$

  $6.605\text{ miles}$

Given:

The coordinates of the points $V,S\text{ and }M$ are given as:

  $V=\left(0,0\right),S=\left(-6,5\right)\text{ and }M=\left(-1.5,8\right)$

Formula Used:

The straight line distance between two points are $(x_1,y_1)$ and $(x_2,y_2)$ is given as:

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

Calculation:

The total distance travelled from $V\text{ to }M$ is the distance from $V\text{ to }S$ and then from $S\text{ to }M.$ Now, the distance between $V\text{ and }S$ is:

  $\begin{array}{l}{d}_1=\sqrt{{\left(-6-0\right)}^2+{\left(5-0\right)}^2}\\ d_1=\sqrt{36+25}\\ d_1=\sqrt{61}\\ d_1=7.81\end{array}$

And, the distance between $S\text{ and }M$ is:

  $\begin{array}{l}{d}_2=\sqrt{{\left(-1.5+6\right)}^2+{\left(8-5\right)}^2}\\ d_2=\sqrt{{\left(4.5\right)}^2+{\left(3\right)}^2}\\ d_2=\sqrt{20.25+9}\\ d_2=\sqrt{29.25}\\ d_2=5.40\end{array}$

Thus, the total distance from $$

  $V\text{ to }M$ is:

  $\begin{array}{l}d=d_1+d_2\\ d=7.81+5.40\\ d=13.21\text{ units}\end{array}$

Here, each unit represents 0.5 miles. Hence, the effective distance travelled from $V\text{ to }M$ is:

  $\begin{array}{l}d=13.21\times 0.5\\ d=6.605\text{ miles}\end{array}$

Hence, the total distance travelled from $V\text{ and }M$ is $6.605\text{ miles}$ .

To determine the distance from $M$ back to $V$ through $P.$

  $8.4\text{ miles}$

Given:

The coordinates of points $M,V$ and $P$ are given as:

  $M=\left(-1.5,8\right),V\left(0,0\right)\text{ and }P\left(3,11\right)$

Formula Given:

The straight line distance between two points are $(x_1,y_1)$ and $(x_2,y_2)$ is given as:

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

Calculation:

Now, the distance from $M\text{ to }P$ is:

  $\begin{array}{l}{d}_1=\sqrt{{\left(3+1.5\right)}^2+{\left(11-8\right)}^2}\\ d_1=\sqrt{{\left(4.5\right)}^2+{\left(3\right)}^2}\\ d_1=\sqrt{20.25+9}\\ d_1=\sqrt{29.25}\\ d_1=5.40\end{array}$

Next, the distance from $P\text{ to }V$ is:

  $\begin{array}{l}{d}_2=\sqrt{{\left(0-3\right)}^2+{\left(0-11\right)}^2}\\ d_2=\sqrt{{\left(3\right)}^2+{\left(11\right)}^2}\\ d_2=\sqrt{9+121}\\ d_2=\sqrt{130}\\ d_2=11.4\end{array}$

Thus, the total distance from $M\text{ to }V$ through $P$ is:

  $\begin{array}{l}d=d_1+d_2\\ d=5.40+11.4\\ d=16.80\text{ units}\end{array}$

Here, each unit represents 0.5 miles. Hence, the effective distance travelled from $M\text{ to }V$ through $P$ is:

  $\begin{array}{l}d=16.80\times 0.5\\ d=8.40\text{ miles}\end{array}$

Hence, the total distance travelled from $V\text{ and }M$ is $8.40\text{ miles}$ .