Related Rates of Change
The Big Idea: Given 2 or more variables whose
measures change with respect to time,
1. Write an equation that relates the variables.
2. Use derivatives with respect to time to see
how their rates of change are related.
Ex. 1 A circular puddle of water is expanding at the rate of 16π
in2 per minute. At what rate is the radius growing when r = 4 in.?
Ex. 2. If the radius of an expanding sphere is growing at
an the rate of 0.1 cm/sec, at what rate is the volume
growing at the moment the radius is 10 cm?
4
V   r3
3
dV
dr
 4 r 2
dt
dt
dV
cm 
2 
 4 10cm    0.1

dt
sec


dV
cm3
 40
dt
sec
The sphere is growing at a rate of 40 cm3 / sec .

Steps for Related Rates Problems:
1. When necessary, draw a picture (sketch).
2. Identify known information.
3. Identify what you are looking for.
4. Write an equation to relate the variables.
5. Differentiate both sides with respect to t.
6. Plug in known quantities & solve.

Water is draining from a cylindrical tank of radius
3 at 3000 cm3/second. How fast is the water
level dropping?
dV
cm3
 3000
dt
sec
V   r 2 h, with r  3
V  9 h
dV
dh
 9
dt
dt
cm3
dh
3000
 9
sec
dt
dh
Find
dt
(We need a formula to
relate V and h. )
dh
1000 cm3

sec
dt
3
The water level is dropping at 1000/3π cm/sec.

Hot Air Balloon Problem:
h
h
tan  
500
d
1 dh
2
sec 

dt 500 dt
2
1 dh
2  0.14  
500 dt
 

500ft
The balloon is rising at a rate of
140 feet per minute.

Truck Problem:
Truck A travels east at 40 mi/hr.
Truck B travels north at 30 mi/hr.
How fast is the distance between the
trucks changing 6 minutes later?
r  t x d y  z
1
1 dz
dx
dy
40
2 x 10  42 y 30 10
2z  3
dt
dt
dt
32  42  z 2
dz
4  40  3  30
2 5
9  16  z
dt
2 dz
25

z
250  5
dz
dt
50 
5 z
dt
2
2
B
2
z
y
dy
 30
dt
A
x
dx
 40
dt
miles
50
hour

A 14 foot ladder is leaning against a wall. If the top of the ladder slips down
the wall at a rate of 2 ft/s, how fast will the bottom be moving away from the
wall when the top is 6 ft above the ground?
dy
 2
dt
x 2  y 2  L2 x2  62  142 x  4 10
y
dL
14 dt  0
L
x
dx
dt
2x

2 4 10

dx
dy
dL
 2y
 2L
dt
dt
dt
dx
 2  6  2   2 14  0 
dt
dx
3

dt
10
The ladder is moving away at a rate of
How fast is the area changing?
3 10
10
A 14 foot ladder is leaning against a wall. If the top of the ladder slips down
the wall at a rate of 2 ft/s, let’s now find how fast the area of the triangle is
changing when the top is 6 ft above the ground?
dy
 2
dt
y
dL
14 dt  0
L
x
dx
3

dt
10
A man 6 ft tall is walking at a rate of 2 ft/s toward a street light 16
ft tall. At what rate is the length of his shadow changing?
6
x

16 x  y
16
6
x
dx
dt
y
dy
 2
dt
6x  6y  16x
6y  10x
dy
dx
6
 10
dt
dt
dx
6  2   10
dt
dx 6

dt
5
The length of his shadow is reducing at a rate of 6/5 ft/s.
A boat whose deck is 10 ft below the level of a dock, is being
drawn in by means of a rope attached to a pulley on the dock.
When the boat is 24 ft away and approaching the dock at ½
ft/sec, how fast is the rope being pulled in?
2
2
2
dx
1
x

y

R

dt
2
2
2
 24   10   R2
x
R  26
dy
y
dx
dy
dR
R
0
2x
 2y
 2R
dR
dt
dt
dt
dt
dt
dR
 1
2  24      2 10  0   2  26 
dt
 2
dR
6

dt
13
The rope is being pulled in at a rate of 6/13 ft/sec.
A pebble is dropped into a still pool and sends out a circular
ripple whose radius increases at a constant rate of 4 ft/s. How
fast is the area of the region enclosed by the ripple increasing at
the end of 8 seconds.
A  r 2
At t = 8, r = (8)(4) = 32
dr
4
dt
dA
dt
dA
dr
 2r
dt
dt
dA
 2  32  4 
dt
dA
 256
dt
The area is increasing at a rate of 256 ft2/sec.
A spherical container is deflated such that its radius decreases at
a constant rate of 10 cm/min. At what rate must air be removed
when the radius is 5 cm?
4 3
V  r
3
dr
 10
dt
dV
dt
dV
2 dr
 4r
dt
dt
dV
2
 4  5   10   1000
dt
Air must be removed at a rate of 1000 ft3/min.
Sand pours into a conical pile whose height is always one half
its diameter. If the height increases at a constant rate of 4
ft/min, at what rate is sand pouring from the chute when the pile
is 15 ft high?
1
1 2
h

d
V  r h
2
3
1
1
3
h   2r 
dh
V


h
4
2
3
dt
hr
dV
2 dh
 h
dt
dt
dV
dt
dV
2
  15   4 
dt
dV
 900
dt
The sand is pouring from the chute at a rate of 900 ft3/min.
Liquid is pouring out of a cone shaped filter at a rate of 3 cubic inches per
minute. Assume that the height of the cone is 12 inches and the radius of
the base of the cone is 3 inches. How rapidly is the depth of the liquid in the
filter decreasing when the level is 6 inches deep?
3
1 2
V  r h
3
2
r
12
h
dV
 3
dt
1 1 
V   h h
3 4 
1
V
h3
48
dV
3
2 dh

h
dt 48
dt
3
2 dh
3 
6
48
dt
4 dh

3 dt
r
h

3 12
1
r h
4
The depth of the
liquid is decreasing
at a rate of 4 in/sec.
3
If xy 2  20 and x is decreasing at the rate of 3 units per second,
the rate at which y is changing when y = 2 is nearest to:
a. –0.6 u/s
b. –0.2 u/s
xy 2  20
x  2   20
x5
2
c. 0.2 u/s
d. 0.6 u/s e. 1.0 u/s
 dx  2  dy 
 dt  y   2y dt   x   0
 


dy 

2
 3  2   2  2    5   0
dt 

dy
20
 12
dt
 
 
dy
3

dt
5
A particle moves along a curve x 2 y  2 at time t  0.
dx
dy
at that time?
 8 when x  1, what is the value of
If
dt
dt
x2y  2
2
 -1 y  2  y  2
 dx 
 dy  2
 2 x dt   y    dt  x  0


 
 


dx 
2

2

1
2

8

1
   dt         0


dx
4
 8
dt
dx
2
dt