   Chapter 11.5, Problem 103E

Chapter
Section
Textbook Problem

# DistanceTwo insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point ( x , y , z ) on the line x = 6 + t , y = 8 − t , z = 3 + t . Also, at time t , the second insect is at the point ( x , y , z ) on the line x = 1 + t , y = 2 + t , z = 2 t . Assume that distances are given in inches.(a) Find the distance between the two insects at time t = 0 .(b) Use a graphing utility to graph the distance between the insects from t = 0 to t = 10 .(c) Using the graph from part (b), what can you conclude about the distance between the insects?(d) How close to each other do the insects get?

(a)

To determine

To calculate: Find the distance between two insects at the point (x,y,z) at t=0.

Explanation

Given:

For time t,

T the first insect is at the point (x,y,z) on the line given by,

x=6+t,y=8t,z=3+t

The second insect is at the point (x,y,z) on the line given by,

x=1+t,y=2+t,z=2t

Formula used:

The distance between two points (x1,y1,z1) and (x2,y2,z2) is given by,

D=(x2x1)2+(y2y1)2+(z2z1)2

Calculation:

Lets find the position of first insect

Put t=0 in the given equation of line. Hence,

x=6+0=6y=80=8z=3+0=3

Therefore, at time t=0, first insect is at point P=(x1,y1,z1)=(6,8,3)

Now Lets find the position of first insect

Put t<

(b)

To determine

To graph: Find the distance between two insects from t=0 to t=10.

(c)

To determine
For the graph from part (b), Interpret the graph

(d)

To determine

To calculate: Find the minimum distance between the insects at the point (x,y,z)

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