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
m3/min. How fast is the water level rising if the base of the tank is a
rectangle of dimensions 2 × 3 meters?
Geometric Diagram:
dh
dV
 ???
 0.3
dt
dt
Geometric Model:
Constants
V  lwh  6h
Implicit Differentiation
dV
dh
dh 1 dV 1
6 

  0.3  0.05 m/min
dt
dt
dt 6 dt 6
Water pours into a conical tank of height 10 m and radius 4 m at a
rate of 6 m3/min.
(a) At what rate is the water level
rising when the level is 5 m high?
r 4
  r  0.4h
dh
 ???
h 10
dt
Geometric Model:
1 2
1
2
Geometric Diagram:
V   r h    0.4h  h
3
3
dV
 0.16  3
2 dh

  0.16   h
 h 
dt
dt
 3 
dh
dh
 0.477 m/min.
6   0.16    25 
dt
dt
Water pours into a conical tank of height 10 m and radius 4 m at a
rate of 6 m3/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 Δh 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?
y
tan    y  6 tan 
6
Geometric Model:
dy
d
2
 6sec 
dt
dt
 6sec
2

3
 0.9   6  0.9  4   21.6 km/min
dy
 ???
dt
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  4.52
c
L  x 2  4.52   6  h 
dL
L is constant 
0
dt
dL 1 2
dx  dh
2 1/ 2 
  x  4.5   2 x  
dt 2
 dt  dt
dx
x
5  3

dh
dt
dh


 2.230 m/s

???
2
2
2
2
dt
x  4.5
5  4.5
dt
Geometric Model:
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
 ???
dt t 1
Constant
5
y  25  x 2
1.5
dx
 0.8
dt
Geometric Model:
Implicit Differentiation
dx
dy
x  y  25  2 x  2 y
0
dt
dt
dx
2 x
  2.25  0.8 
dy
dt


 0.403 m/s
2
dt
2y
25  2.25
2
2