Chapter 1, Problem 6PS

To determine

To find: the coordinates of the point $\left(x,y\right)$ Then write an equation for the path of the ball.

Answer to Problem 6PS

The piecewise equation of path of the ball is given by

  $\{\begin{array}{l}\frac{12}{7}x-\frac{16}{7},2.2\le x\le 6\\ -\frac{12}{7}x+\frac{128}{7},6\le x\le 9.5\end{array}$

Explanation of Solution

Given:

A golfer is trying to make a hole-in-one on the miniature golf green shown. Then golf ball is at point $\left(2.5,2\right)$ and the hole is at the point $\left(9.5,2\right)$ . The golfer wants to bank the ball off the side of the wall of the green at the point $\left(x,y\right)$ .

  

Calculation:

From the diagram we can see that y-coordinate is at 8 and x-coordinate is midpoint of gold ball and hole.

  $\left(x,y\right)=\left(\frac{2.5+9.5}{2},8\right)=\left(6,8\right)$

Now through two known points, the equation of line passing through these points can be found

To find equation of line passing through $\left(2.5,2\right)\&\left(6,8\right)$ first find the slope of the line using

  $\begin{array}{l}y=\frac{y_2-y_1}{x_2-x_1}\\ y=\frac{8-2}{6-2.5}\\ y=\frac{6}{3.5}\\ y=\frac{12}{7}\end{array}$

Now use point slope form $\left(y=mx+c\right)$ of a line to write the equation of line.

  $\begin{array}{l}8=\frac{12}{7}\times 6+c\\ c=-\frac{16}{7}\end{array}$

So, $y=\frac{12}{7}x-\frac{16}{7}$

To find equation of line passing through $\left(9.5,2\right)\&\left(6,8\right)$ first find the slope of the line using

  $\begin{array}{l}y=\frac{y_2-y_1}{x_2-x_1}\\ y=\frac{8-2}{6-9.5}\\ y=-\frac{6}{3.5}\\ y=-\frac{12}{7}\end{array}$

Now use point slope form $\left(y=mx+c\right)$ of a line to write the equation of line.

  $\begin{array}{l}8=-\frac{12}{7}\times 6+c\\ c=\frac{128}{7}\end{array}$

So $y=-\frac{12}{7}x+\frac{128}{7}$

So, piecewise equation of path of the ball is given by

  $\{\begin{array}{l}\frac{12}{7}x-\frac{16}{7},2.2\le x\le 6\\ -\frac{12}{7}x+\frac{128}{7},6\le x\le 9.5\end{array}$