Chapter 9, Problem 19SGR

(a).

To determine

To find: The how far is the waterfall.

Answer to Problem 19SGR

The waterfall is approximately $9.434$ miles away from the camp.

Explanation of Solution

Given information:

A waterfall is situated $5$ miles south and $8$ miles east of the camp of Marc.

Calculation:

A waterfall is situated $5$ miles south and $8$ miles east of the camp of Marc.

  

From the figure, O represents the location of the camp and point B represents the location of the waterfall.

Co-ordinates of point O will be $\left(0,0\right)$ and of point B will be $\left(8,-5\right)$ Consider Distance formula of the two points with co-ordinates $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is as follows

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

Substitute $\left(x_1,y_1\right)=\left(0,0\right)$ and $\left(x_2,y_2\right)=\left(8,-5\right)$ the distance between waterfall and camp be

  $\begin{array}{l}d=\sqrt{{\left(8-0\right)}^2+{\left(-5-0\right)}^2}\\ d=\sqrt{89}\\ d=9.4339811\end{array}$

Approximation gives

  $d\approx 9.434$

Therefore, the waterfall is approximately $9.434$ miles away from the camp.

(b).

To determine

To find: The where Marc should stop for lunch halfway to the waterfall.

Answer to Problem 19SGR

TheMarc should stop for lunch at point $\left(4,-2.5\right)$ .

Explanation of Solution

Given information:

A waterfall is situated $5$ miles south and $8$ miles east of the camp of Marc.

Calculation:

Marc wants to stop for lunch halfway to the waterfall. Compute the co-ordinates of the mid-points of camp and waterfall using Mid-point formula as follows:

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

Substitute $\left(x_1,y_1\right)=\left(0,0\right)$ and $\left(x_2,y_2\right)=\left(8,-5\right)$ the co-ordinates of the mid-points of camp will be,

  $\begin{array}{c}\left(\frac{x_2+x_1}{2},\frac{y_2+y_1}{2}\right)=\left(\frac{0+8}{2},\frac{0+(-5)}{2}\right)\\ =\left(4,-2.5\right)\end{array}$

Therefore, the Marc should stop for lunch at point $\left(4,-2.5\right)$ .