Transcript related-rate
In related-rate problems, the goal is to calculate an unknown rate of change in terms of other rates of change that are known. The “sliding ladder problem” is a good example: A ladder leans against a wall as the bottom is pulled away at constant velocity. How fast does the top of the ladder move? What is interesting and perhaps surprising is that the top and bottom travel at different speeds. Figure 1 shows this clearly: The bottom travels the same distance over each time interval, but the top travels farther during the second time interval than the first. In other words, the top is speeding up while the bottom moves at a constant speed. In a moment, we use calculus to find the velocity of the ladder’s top.
Filling a Rectangular Tank Water pours into a fish tank at a rate of 0.3
m 3
/min. How fast is the water level rising if the base of the tank is a rectangle of dimensions 2 × 3 meters? Geometric Diagram:
dV
0.3
dt dh
???
dt
Geometric Model:
V
lwh
6
h
Implicit Differentiation
dV dt
6
dh dt
dh dt
1 6
dV dt
1 6 Constants
0.05
m/min
Water pours into a conical tank of height 10 m and radius 4 m at a rate of 6 m 3 /min.
(a) At what rate is the water level rising when the level is 5 m high?
r
4 0.4
h h
10
Geometric Model:
V
1 3 2
r h
1 3
0.4
h
2
h dh
???
dt
Geometric Diagram: 0.16
3 6
h
3 0.16
dV dt
25
0.16
h
2
dh dt dh dt dh
0.477 m/min.
dt
Water pours into a conical tank of height 10 m and radius 4 m at a rate of 6 m 3 /min.
(a) At what rate is the water level rising when the level is 5 m high?
dh
0.477 m/min.
dt
(b) As time passes, what happens to the rate at which the water level rises?
As
h
increases, the water level rises more slowly. This is reasonable if you consider that a thin slice of the cone of width Δ
h
has more volume when
h
is large, so more water is needed to raise the level when
h
is large.
Tracking a Rocket
A spy uses a telescope to track a rocket launched vertically from a launching pad 6 km away. At a certain moment, the angle between the telescope and the ground is equal to and is changing at a rate of 0.9 rad/min. What is the rocket’s velocity at that moment?
tan
y
6
y
6 tan
Geometric Model:
dy
???
dt dy
6 sec 2
dt
6 sec 2 3
d
dt
21.6 km/min
Farmer John’s tractor, traveling at 3 m/s, pulls a rope attached to a bale of hay through a pulley. With dimensions as indicated in the diagram, how fast is the bale rising when the tractor is 5 m from the bale?
c
x
2 4.5
2
L
x
2 4.5
2
Geometric Model:
dL
dt
1 2
x
2 4.5
2 1/ 2
dh
dt x
2
dx x
dt
4.5
2
h
2
x dx dt c
L
is constant
dL dt dh dt
0 5 2 4.5
2
2.230
m/s
dh
???
dt
Sliding Ladder Problem A 5-meter ladder leans against a wall. The bottom of the ladder is 1.5 meters from the wall at time t = 0 and slides away from the wall at a rate of 0.8 m/s. Find the velocity of the top of the ladder at time t = 1.
Geometric Diagram:
dy
???
Constant
dt t
1
5
y
25
x
2
dx
1.5
0.8
dt
Geometric Model:
Implicit Differentiation
x
2
y
2 25 2
x dx dt
2
y dy dt
0
dy
dt
2
x dx dt
2
y
2 0.403 m/s