Chapter 8.6, Problem 37E

a.

To determine

To find: the parametric equations for the path of target drone and the missile.

Answer to Problem 37E

  $\begin{array}{l}\text{Drone: }x=3-t\text{ and }y=4\\ \text{Missile: }x=2+t\text{ and }y=2+2t\end{array}$

Explanation of Solution

Given information:

Vector equation for path of target drone $\left(x,y\right)=\left(3,4\right)+t\left(-1,0\right)$

The trajectory represented by the missile $\left(x,y\right)=\left(2,2\right)+t\left(1,2\right)$

Calculation:

The parametric equations for targeted drone can be evaluated as follows:

  $\begin{array}{l}\left(x,y\right)=\left(3,4\right)+t\left(-1,0\right)\\ \begin{array}{ccc}\begin{array}{l}x=3+\left(-1\right)t\\ x=3-t\end{array}& \begin{array}{l}\text{and}\\ \text{and}\end{array}& \begin{array}{l}y=4+0t\\ y=4\end{array}\end{array}\end{array}$

Thus, the parametric equations for drone are $x=3-t\text{ and }y=4$ .

The parametric equations for missile can be evaluated as follows:

  $\begin{array}{l}\left(x,y\right)=\left(2,2\right)+t\left(1,2\right)\\ \begin{array}{ccc}x=2+t& \text{and}& y=2+2t\end{array}\end{array}$

Thus, the parametric equation for missile are $x=2+t\text{ and }y=2+2t$ .

b.

To determine

To state: whether the missile intercepts the target drone.

Answer to Problem 37E

No

Explanation of Solution

Given information:

Vector equation for path of target drone $\left(x,y\right)=\left(3,4\right)+t\left(-1,0\right)$

The trajectory represented by the missile $\left(x,y\right)=\left(2,2\right)+t\left(1,2\right)$

The parametric equations evaluated in part a. are shown below.

  $\begin{array}{l}\text{Drone: }x=3-t\text{ and }y=4\\ \text{Missile: }x=2+t\text{ and }y=2+2t\end{array}$

Calculation:

Compare the parametric equations of both drone and missile for $x$ and evaluate $t$ as shown below.

  $\begin{array}{c}3-t=2+t\\ -2t=-1\\ t=\frac{1}{2}\end{array}$

At $t=\frac{1}{2}$ , the missile has a $y$ -coordinate of 3 which is not equal to the $y$ coordinate of drone. Thus, the missile does not intercept the target drone.