Chapter 8.6, Problem 11CFU

a.

To determine

To find:parametric equations for the path of each player.

Answer to Problem 11CFU

Parametric equations for receiver:

  $\begin{array}{c}x=5\\ y=50-10t\end{array}$

Parametric equations for defensive player:

  $\begin{array}{l}x=10-0.9t\\ y=54-10.72t\end{array}$

Explanation of Solution

Given information:

The equations for the receiver and the defensive player are shown below.

  $\begin{array}{c}\left(x-5,y-50\right)=t\left(0,-10\right)\\ \left(x-10,y-54\right)=t\left(-0.9,-10.72\right)\end{array}$

Calculation:

The parametric equation for receiver will be evaluated as follows:

  $\begin{array}{c}\left(x-5,y-50\right)=t\left(0,-10\right)\\ \left(x-5,y-50\right)=\left(0,-10t\right)\\ x-5=0\\ x=5\\ \\ y-50=-10t\\ y=50-10t\end{array}$

Thus, the parametric equations for the receiver are $x=5\text{ and }y=50-10t$ .

The parametric equation for defensive player will be evaluated as follows:

  $\begin{array}{c}\left(x-10,y-54\right)=t\left(-0.9,-10.72\right)\\ \left(x-10,y-54\right)=\left(-0.9t,-10.72t\right)\\ x-10=-0.9t\\ x=10-0.9t\\ \\ y-54=-10.72t\\ y=54-10.72t\end{array}$

Thus, the parametric equations for the defensive player are $x=10-0.9t\text{ and }y=54-10.72t$ .

b.

To determine

To state:whether the player will reach the goal line before the defensive player catches him.

Answer to Problem 11CFU

Yes

Explanation of Solution

Given information:

The receiver catches the ball on line $y=50$ .

The defensive player goal line is $y=0$ .

Parametric equations for receiver:

  $\begin{array}{c}x=5\\ y=50-10t\end{array}$

Parametric equations for defensive player:

  $\begin{array}{l}x=10-0.9t\\ y=54-10.72t\end{array}$

Calculation:

Evaluate the value of $t$ for the situation when receiver has $y=0$ as shown below.

  $\begin{array}{c}y=50-10t\\ 0=50-10t\\ 10t=50\\ t=5\end{array}$

Put $t=5$ for the equation of defensive player and evaluate $\left(x,y\right)$ as shown below.

  $\begin{array}{c}x=10-0.9t\\ =10-0.9\left(5\right)\\ =10-4.5\\ =5.5\\ \\ y=54-10.72t\\ =54-10.72\left(5\right)\\ =0.4\end{array}$

Since the point value of $y\ne 0$ for the defensive player. Thus, the player will reach the goal line before the defensive player catches him.