Chapter ISG, Problem 9.7.14P

a.

To determine

To graph: The path of the rubber bands and the path of the target on the coordinate plane at the right.

Explanation of Solution

Given Information: Rubber bands follows a path along $y=\frac{1}{2}x+2$ whereas target follows a path along $y=-x^2-2x+4$ .

Graph:

  

b.

To determine

To find: The point(s) at which player hits the target.

Answer to Problem 9.7.14P

Player can hit the target at either at $(-3.137,0.431)or(0.637,2.319)$ .

Explanation of Solution

Given Information: Rubber bands follows a path along $y=\frac{1}{2}x+2$ whereas target follows a path along $y=-x^2-2x+4$ .

Calculation:

For player to hit the target, paths should intersect,

  $\begin{array}{l}-x^2-2x+4=\frac{1}{2}x+2\\ \\ -x^2-2x-\frac{1}{2}x+2=0\\ \\ -x^2-\frac{5}{2}x+2=0\end{array}$

Multiplying both sides by $-1$ ,

  $x^2+\frac{5}{2}x-2=0$

  $\begin{array}{l}D=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ \\ =\frac{-\frac{5}{2}\pm \sqrt{{\left(\frac{5}{2}\right)}^2-4(1)(-2)}}{2(1)}\\ \\ =\frac{-\frac{5}{2}\pm \sqrt{\frac{25}{4}+\frac{8}{1}}}{2}\\ \\ =\frac{-2.5\pm 3.7749}{2}\end{array}$

  $\Rightarrow x=0.637458or-3.13745$

  $\begin{array}{l}\text{For x=-0}\text{.637458,}\\ \\ y=\frac{1}{2}x+2\\ \\ =\frac{1}{2}(0.637458)+2\\ \\ =2.318729\\ \\ \text{For x=-3}\text{.13745,}\\ \\ y=\frac{1}{2}x+2\\ \\ =\frac{1}{2}(-3.13745)+2\\ \\ =0.431275\end{array}$

Thus, player can hit the target at either at $(-3.137,0.431)or(0.637,2.319)$ .

c.

To determine

To calculate: The coordinates at which rubber bands may hit the target and then to compare them with those calculated in part $b$ .

Answer to Problem 9.7.14P

Calculator outputted that coordinates are $(0.6,2.3)and(-3.1,0.4)$, which is same as the result from part $b$ .

Explanation of Solution

Given Information: Calculator should be used.

Calculation:

An online calculator displayed the following results,

  

  $\begin{array}{l}\frac{1}{2}(0.6374586088176875)+2=2.31872930441\\ \\ \frac{1}{2}(-3.1374586088176875)+2=0.43127069559\end{array}$

Rounding the coordinates, result from calculator is $(0.6,2.3)and(-3.1,0.4)$.

Thus, calculator outputted that coordinates are $(0.6,2.3)and(-3.1,0.4)$, which is same as the result from part $b$ .