Chapter 10.8, Problem 42E

(a)

To determine

To graph: The equations on the same axis.

Answer to Problem 42E

The graph

Explanation of Solution

Given information: Entertainment: In a science fiction movie, astronomers track a large incoming asteroid and predict that it will strike Earth with disastrous results. Suppose certain latitude of Earth’s surface is modeled byy1y3 $x^2+y^2=40$ and the path of the asteroid is modeled by

  $x=0.25y^2+5$ .

  

Calculation:

The graph:

  

(b)

To determine

To find: If the asteroid strikes Earth. If so, find the coordinates of the point of impact.

Answer to Problem 42E

  $\left(6,2\right)\text{or}\left(6,-2\right),\text{ depending on which direction the asteroid is coming from}$

Explanation of Solution

Given information: Entertainment: In a science fiction movie, astronomers track a large incoming asteroid and predict that it will strike Earth with disastrous results. Suppose certain latitude of Earth’s surface is modeled by $x^2+y^2=40$ and the path of the asteroid is modeled by $x=0.25y^2+5$ .

  

Calculation:

Solve for $y^2$ :

  $\begin{array}{l}{y}^2=40-x^2\\ y^2=4\left(x-5\right)\end{array}$

Equate and simplify:

  $\begin{array}{l}40-x^2=4\left(x-5\right)\\ 40-x^2=4x-20\\ x^2+4x-60=0\\ x=\frac{-4\pm 16}{2}=6\text{ or }-10\end{array}$

  $x=-10\text{ is }$ not in the asteroid’s path, so that leaves us with x = 6. Plugging that into the first equation:

  $y^2=40-6^2=4\to y=\pm 2$

Thus,

  $\left(6,2\right)\text{or}\left(6,-2\right),\text{ depending on which direction the asteroid is coming from}$

(c)

To determine

To describe: This situation with parametric equations. Assume both the asteroid and Earth’s surface are moving counterclockwise.

Answer to Problem 42E

  $\begin{array}{l}\text{Earth:}\\ x=2\sqrt{10}\cos t,y=2\sqrt{10}\sin t\\ \text{Astroid:}\\ x=t^2+5\text{, }y=2t\end{array}$

Explanation of Solution

Given information: Entertainment: In a science fiction movie, astronomers track a large incoming asteroid and predict that it will strike Earth with disastrous results. Suppose certain latitude of Earth’s surface is modeled by $x^2+y^2=40$ and the path of the asteroid is modeled by $x=0.25y^2+5$ .

  

Calculation:

The parametric equations of a circle have the form

  $x=r\cos t,y=r\sin t$

Here, $r=\sqrt{40}=2\sqrt{10,}$ and so,

  $x=2\sqrt{10}\cos t,y=2\sqrt{10}\sin t$

For the parabola, rearrange the terms so that the independent term is next to the non-squared variable:

  $x-5=\frac{y^2}{4}$

Equate both sides to $t^2$ :

  $\begin{array}{l}x-5=t^2\to x=t^2+5\\ \frac{y^2}{4}=t^2\to y=\pm 2t\end{array}$

The sign of $2t$ gives the direction of the parabola

We’ll choose the positive sign to make it counterclockwise.

  $x=t^2+5,y=2t$

Thus,

  $\begin{array}{l}\text{Earth:}\\ x=2\sqrt{10}\cos t,y=2\sqrt{10}\sin t\\ \text{Astroid:}\\ x=t^2+5\text{, }y=2t\end{array}$

(d)

To determine

To graph: The equations found in part c using a graphing calculator. Use a window that shows complete graphs of both Earth’s surface and the asteroid’s path.

Answer to Problem 42E

  $\begin{array}{l}\text{Earth:}\\ x=2\sqrt{10}\cos t,y=2\sqrt{10}\sin t\\ \text{Astroid:}\\ x=t^2+5\text{, }y=2t\end{array}$

Explanation of Solution

Given information: Entertainment: In a science fiction movie, astronomers track a large incoming asteroid and predict that it will strike Earth with disastrous results. Suppose certain latitude of Earth’s surface is modeled by $x^2+y^2=40$ and the path of the asteroid is modeled by $x=0.25y^2+5$ .

  

Calculation: