Answers and Explanations 287
Answers
401–500
Taking the derivative of both sides with respect to time gives you
2
2
2
3
2
3
1 3
2 3
D dD
dt
x dx
dt
x
dx
dt
x
dx
dt
=
+
+
−
−
From the given values x = 8 centimeters and y = 3 centimeters, you find that the
distance from the origin is D = 73 centimeters. Use these values along with dx
dt
= 5
centimeters per second and solve for dD
dt
:
2 73
2 8 5 2
3
8
5 2
3
8
5
4 83
1 3
2 3
dD
dt
dD
dt
=
+
( +
≈
−
−
( )( )
( )
)
( ) ( )
. cm/s
441.
2.95 mi/h
Let x be the distance that the person who is walking west has traveled, let y be the
distance that the person traveling southwest has traveled, and let z be the distance
between them.
Use the law of cosines to relate x, y, and z:
z
x
y
xy
z
x
y
xy
2
2
2
2
2
2
2
4 5
2
=
+ −
°
=
+ −
cos(
)
Take the derivative of both sides of the equation with respect to time:
2
2
2
2
z dz
dt
x dx
dt
y
dy
dt
dx
dt
y x
dy
dt
=
+
−
( ) +

 

 
After 40 minutes, or 2
3
hour, x = 4
3
and y = 8
3
. Using the initial equation, you find that z
≈ 1.96 miles at this time. Substitute these values into the derivative and solve for dz
dt
:
2 1 96
2 4
3
2 2 8
3
4
2 2 8
3
4
3
4
2 9
( . )
()
( )
( )
.
dz
dt
dz
dt
=
( ) +
−
+
≈
( ) ( )
( )

 

 
5 5 mi/h
After 40 minutes, the distance between these people is increasing at a rate of about
2.95 miles per hour.