Chapter 4.2, Problem 76E

Solution

To identify: The where along the portion CD should you direct ball A so that it will bounce off CD and strike ball B. Assume that A obeys the law of reflection relative to rail CD.

  

Ball A should be directed at 18 in. along the portion of CD of the railing so that it will bounce off CD and strike the ball B.

Given information

  

Explanation:

Consider the given information

The adjacent side of each triangle created are labeled as $xandy$ Another thing that we can get from the diagram is that since we assume that ball A obeys the law of reflection, the angle of incidence is equal to the angle when the ball bounced off CD to strike ball B.

Since the lengths of the opposite sides are given, and we want to find a function for the adjacent sides (labeled as $xandy$ ), we must identify a trigonometric function that involves the two.

  $\tan \theta =\frac{\text{opp}\text{.}}{\text{adj}\text{.}}$

The following functions are created when we express the distances $x$

The following functions are created when we express the distances $xandy$ as a function of the angle $\theta$ .

  $\tan \theta =\frac{15}{x}\tan \theta =\frac{10}{y}$

Since we know that the angle $\theta$ are the same, then we created a single equation by equating the 2 functions from step 3. We have

  $\begin{array}{c}\frac{15}{x}=\frac{10}{y}\\ 15y=10x\end{array}$

From the given, we know that $x+y=30$ . Manipulating this equation, we were able to create an equation to solve for y.

  $y=30-x$

Substitute this on the equation from step 4 to solve for $x$ .

  $\begin{array}{c}15y=10x\\ 15\left(30-x\right)=10x\\ 450-15x=10x\end{array}$

Combine like terms then to get the value of $x$

  $\begin{array}{c}450=25x\\ x=\frac{450}{25}\\ x=18\end{array}$

Hence, ball A should be directed at 18 in. along the portion of CD of the railing so that it will bounce off CD and strike the ball B.