Related Rates - (2.8)
What are related rates?
Example Suppose a bug is crawling around the circle of radius 5. The coordinates of the location where
dy
the bug is at the time are functions of t : xŸt , yŸt . If we know,
" 6 units/second when
dt
x 3 and y 0, what is dx at this moment?
dt
We know the relation of x and y : x 2 y 2 25 (the equation of the circle with radius 5 .
Consider both x and y are functions of time:
x xŸt and y yŸt dy
is:
The relation of the rates of change of x and y : dx and
dt
dt
dy
d Ÿx 2 y 2 d Ÿ25 ,
2x dx 2y
0
dt
dt
dt
dt
dx " y dy , when x 3, y 25 " 9 4, dx " 4 Ÿ"6 8 units/second
x dt
3
dt
dt
At the location 3, 4 , the value of y is decreasing and the value of x is increasing. So, the bug is moving
around the circle clockwise.
Example If we are pumping air into a balloon, both the volume and the radius of the balloon are
increasing with respect to time t. Let VŸt and rŸt be the volume and the radius of the balloon at
the time t. Find the relation of the relative rates dV and dr . If we know the rate of change of r is
dt
dt
0. 25 inch/second when r 3 inches, what is the rate of change of the volume at the moment?
We know the relation of V and r : V 4 =r 3 . Implicitly, V VŸt and r rŸt .
3
dV |
dV 4=r 2 dr ,
4=Ÿ3 2 Ÿ0. 25 28. 274 33 cubic inches/second
dr
dt r3, dt 0.25
dt
dt
Example At 10:00am a fishing boat leaves the dock and heads due south at 20 mph. At the same moment
a ferry located 60 miles directly west of the dock is traveling toward the dock at 25 mph. Assume
that both the fishing boat and the ferry maintain their speeds for the next two hours. How fast is the
distance between the boats changing at 11:00am? Are the boats getting farther apart or closer
together?
Let yŸt be the distance that the fishing boat travels t hours after 10:00am, and let xŸt be the distance that the
ferry travels t hours after 10:00am. Let DŸt be the distance between the boat and the ferry t hours after 10:00am.
We want to find dD when t 1 hour.
dt
1
dy
20 mph, dx 25 mph
dt
dt
2
¡DŸt ¢ Ÿ60 " xŸt 2 ¡yŸt ¢ 2
When t 1, xŸ1 25, yŸ1 20
DŸ1 Ÿ60 " 25 2 20 2 5 65
40. 311 29 miles
2DŸt dD 2Ÿ60 " xŸt " dx
dt
dt
When t 1,
2yŸt dy
dt
dy
dD 1
Ÿ60 " xŸ1 " dx 2yŸ1 dt
dt
dt
DŸ1 1 ŸŸ60 " 25 Ÿ"25 Ÿ20 Ÿ20 5 65
"11. 783 30 mph
Example A ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from the
wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the
ladder is 6 ft from the wall.
Let xŸt be the distance from the bottom of the ladder to the wall and let yŸt be the distance from the top of
the ladder to the ground.
Relations of xŸt and yŸt : x 2 y 2 100
dx 1 foot/second
dt
dy
when x 6 ft, and dx 1.
Want to find
dt
dx
d ¡x 2 y 2 ¢ d Ÿ100 dt
dt
dy
dy
2x dx 2y
0,
" xy dx
dt
dt
dt
dt
dy
x
x
dx
dx
"y
"
dt
dt
100 " x 2 dt
6
Ÿ1 " 68 " 34 ft/s
"
100 " 36
Example A water tank has the shape of an inverted circular cone with base radius 2m and height 4m. If
water is being pumped into the tank at a rate of 2m 3 /min, find the rate at which the water level is
rising when the water is 3 m deep.
2
Let V, r, and h be the volume of the water,
the radius of the surface, and the height at time t.
We know: r 2, h 4, dV 2 m 3 /min.
dt
dh
when h 3.
Want to find:
dt
Relation of r and h : r 2 1 , r 1 h.
2
4
2
h
Relation of V and h :
2
= h h = h3
12
2
Relation of dV and dh :
dt
dt
= h 3 = h 2 dh ,
d ŸV d
4
dt
dt
dt 12
dh 4 dV
dt
=h 2 dt
4 Ÿ2 8 0. 282 94 m/min
dh 9=
dt
=Ÿ3 2
V
1
3
=r 2 h 1
3
Example A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole
with a speed of 5 ft/s along a straight path. How fast is the length of his shadow changing when he
is 40 ft from the pole? Does the change of the length of his shadow depend on how far he is away
from the pole?
Let xŸt be the distance from between the man and the street light, and yŸt be the length of the his shadow at
the time t.
We know: dx 5 feet/second
dt
dy
when x 40.
Want to find:
dt
Relation of x and y :
y
xy
, 6x 6y 15y, 6x 9y, y 2 x
6
3
15
dy
dy
:
Relation of dx and
2 dx
3 dt
dt
dt
dt
dy
23 Ÿ5 103 feet/second
dt
dy
is a constant, that is it does not depend
Note that
dt
on x, how far the man is away from the street light.
3