Related Rates
I. Procedure
A.) State what is given and what is to be found! Draw a
diagram; introduce variables with quantities that can
change and constants that do not change. Denote
DECREASE with a negative sign.
B.) State the EQUATION, valid at any time t, among
the variables involved.
C.) Implicitly differentiate the equation with respect to
time in order to obtain a relationship about the RATES
of the variables valid at any time t.
D.) NOW, and NOT BEFORE, introduce specific
value(s) which the variables take at the instant in the
equation.
E.) Answer the question and include proper units in the
answer.
II. Examples
A.) A 12 foot ladder stands against a vertical wall. If the
lower end of the ladder slips away from the wall at a
rate of 2 ft./sec., how fast is the top of the ladder
coming down the wall when the ladder is 4 feet above
the ground?
12
y
dy
?
dt
x
dx
2
dt
dy
dx
Set up: Find
when
2 & y4
dt
dt
12  x  y
2
2
8 2x
2
dx
dy
0  2x  2 y
dt
dt
dy
0  2x  2  2  4
dt
144  x 2  42


dy
0  2 8 2  2  2  4
dt
dy
4 2 ft./sec. 
dt
B.) A conical reservoir 12 feet deep and 12 feet across
the top is being filled with water at a rate of 5 cubic
feet per minute. How fast is the water level rising
when the water is 4 feet deep?
Diagram:
6
12
r
h
r h

6 12
1
r h
2
h  2r
Set up:
dh
dV
Find
when
5 & h4
dt
dt
1
2 dh
5    4
4
dt
1 2
V  r h
3
2
1 1 
V    h h
3 2 
1
V   h3
12
dV 1 2 dh
 h
dt 4
dt
dh
5  4
dt
dh 5

ft./min.
dt 4
C.) A stone is dropped into a small pond. Concentric
circular ripples spread out and the radius of the
disturbed region increases at a rate of 16 cm/sec. At
what rate does the area of the disturbed region
increase when its radius is 4 cm?
Diagram:
r
dr
 16
dt
Set up:
dA
dr
Find
when
 16 & r  4
dt
dt
A   r2
dA
dr
 2 r
dt
dt
dA
 2  4 16 
dt
dA
 128 cm 2 /sec
dt
D.) Two roads meet at a right angle. Two cars leave the
intersection at the same instant. One car travels
directly north at a speed of 45 mph, the other travels
directly west at a speed of 60 mph. After 25 minutes,
how fast is the distance between them increasing?
Diagram:
C
dx
 60
dt
dy
 45
dt
Set up:
dC
dx
dy
Find
when
 60,
 45 & t  25
dt
dt
dt
C x y
2
2
2
dC
dx
dy
2C
 2x  2 y
dt
dt
dt
dC
dx
dy
C
x y
dt
dt
dt
C 2  18.752  252
31.25  C
dC
31.25
 25  60   18.75  45 
dt
dC
 75 mph
dt
E.) A spherical balloon is deflating so that its radius is
decreasing at the of 2 in./sec. How fast is the volume
of the balloon decreasing when the volume is 288π
in.3?
dV
dr
Set up: Find
when
 2 & V  288
dt
dt
4 3
V  r
3
dV
2 dr
 4 r
dt
dt
4 3
288   r
3
6r
dV
2
 4  6   2 
dt
dV
3
 288 in /sec
dt
F.) A Ferris wheel with a radius of 25 feet is revolving at
the rate of 10 radians per minute. How fast is a
passenger falling when the passenger is 15 feet higher
than the center of the Ferris wheel and the passenger
is on his way back down?
Diagram:
25

dy
?
dt
d
 10
dt
Set up:
dy
d
Find
when
 10 & y  15
dt
dt
y
sin  
25
d
1 dy
cos 

dt
25 dt
1 dy
4
   10  
25 dt
5
dy
 200 ft/min
dt