Chapter 11, Problem 16PS

a.

To determine

To find the distance between the two insects at time $t=0$ .

Answer to Problem 16PS

  $\boxed{8.37in}$

Explanation of Solution

Given information:

Two insects are crawling along different lines in three-space. At time $t$ (in minutes), the first insect is at the point $\left(x,y,z\right)$ on the line given by

  $x=6+t,y=8-t,z=3+t$

Also, at time $t$ the second insect is at the point $\left(x,y,z\right)$ on the line given by

  $x=1+t,y=2+t,z=2t$

Find the distance between the two insects at time $t=0$ .

Calculation:

Here, the distance $d_{AB}$ between two points $A\left(a_x,y_x,z_x\right)$ and $B\left(a_x,y_x,z_x\right)$ is equal to the:

  $d_{AB}=\sqrt{{\left(b_x-a_x\right)}^2+{\left(b_y-a_y\right)}^2+{\left(b_z-a_z\right)}^2}$

Now, at time $t$ the position of the insect $1$ is $\left(6+t,8-t,3+t\right)$ and position of insect $2$ is $\left(1+t,2+t,2t\right)$ .

Thus, the distance between two insects will be:

  $d=\sqrt{{\left(1+t-6-t\right)}^2+{\left(2+t-8+t\right)}^2+{\left(2t-3-t\right)}^2}$

  $=\sqrt{25+{\left(-6+2t\right)}^2+{\left(t-3\right)}^2}$

At $t=0$ ,

  $d=\sqrt{25+36+9}$

  $=8.37$

Hence, distance between two insects at $t=0$ is $\boxed{8.37in}$.

b.

To determine

To graph the distance between the insects by using graphing utility.

Answer to Problem 16PS

  

Explanation of Solution

Given information:

Two insects are crawling along different lines in three-space. At time $t$ (in minutes), the first insect is at the point $\left(x,y,z\right)$ on the line given by

  $x=6+t,y=8-t,z=3+t$

Also, at time $t$ the second insect is at the point $\left(x,y,z\right)$ on the line given by

  $x=1+t,y=2+t,z=2t$

Use a graphing utility to graph the distance between the insects from $t=0$ to $t=10$ .

Calculation:

Here we will use $TI-83$ to graph the distance between two insects.

Now, press $\boxed{Y=}$ button and write the equation:

  

Now, press $\boxed{WINDOW}$ button and make such readings:

  

Now, press $\boxed{GRAPH}$ and graph is like:

  

Hence, the result is shown in the graph.

c.

To determine

To determine the conclusion about the distance between the insects.

Answer to Problem 16PS

  $\boxed{Increasing}$

Explanation of Solution

Given information:

Two insects are crawling along different lines in three-space. At time $t$ (in minutes), the first insect is at the point $\left(x,y,z\right)$ on the line given by

  $x=6+t,y=8-t,z=3+t$

Also, at time $t$ the second insect is at the point $\left(x,y,z\right)$ on the line given by

  $x=1+t,y=2+t,z=2t$ .

Using the graph from part (b), what can you conclude about the distance between the insects?

Calculation:

Hence, from the above graph we have observed that as the time increases, initially the distance decreases but after a certain time it starts $\boxed{Increasing}$.

d.

To determine

To determine how close the insects get to each other.

Answer to Problem 16PS

  $\boxed{Cannotbecalculated}$

Explanation of Solution

Given information:

Two insects are crawling along different lines in three-space. At time $t$ (in minutes), the first insect is at the point $\left(x,y,z\right)$ on the line given by

  $x=6+t,y=8-t,z=3+t$

Also, at time $t$ the second insect is at the point $\left(x,y,z\right)$ on the line given by

  $x=1+t,y=2+t,z=2t$ .

Using the graph from part (b), determine how close the insects get to each other.

Calculation:

Here the distance between the two insects will minimum then the slope of the function becomes zero.

Thus,

  $\begin{array}{l}\frac{d}{dt}\left(d\right)=0\\ \frac{d}{dt}\left(\sqrt{5t^2-30t+70}\right)=0\\ 5t^2-30t+70=0\end{array}$

Now, use general quadratic roots of equation $a{x}^2+bx+c=0$ :

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Thus, on solving $t=3\pm \sqrt{-5}$ only positive value of $t$ is valid. So, the time instant at which the distance is:

  $\begin{array}{l}{d}_{\min }=\sqrt{5{\left(3+\sqrt{-5}\right)}^2-30\left(3+\sqrt{-5}\right)+70}\\ d_{\min }=\sqrt{2\left(9-5+6\sqrt{-5}\right)-90-30\sqrt{-5}+70}\\ d_{\min }=\sqrt{8+12\sqrt{-5}-90-30\sqrt{-5}+70}\\ d_{\min }=\sqrt{-12-18\sqrt{-5}}\end{array}$

Hence, the minimum distance is an imaginary value. Thus, we cannot calculate the minimum distance between the two insects.