Related Rates
3.7
Finding a rate of change that cannot be easily measured by using another rate
that can be is called a Related Rate problem.
Steps for Related Rates Problems:
1.
Draw a picture (sketch).
2. Write down known information.
3. Write down what you are looking for.
4. Write an equation to relate the variables.
5. Differentiate both sides with respect to t.
6. Evaluate.
A ladder, 10 ft tall rests against a wall. If the ladder is sliding away from
the bottom of the wall at 1 ft/sec, how fast is the top of the ladder coming
down the wall when the bottom is 6 ft from the wall?
dx
ft
1
dt
sec
10
y
x
We want
dy/dt when x
=6
x 2  y 2  100
dx
dy
2x  2 y
0
dt
dt
dy  x dx


dt
y dt
At x = 6,
y = 8 by
Pythagorea
n theorem
dy  6
 3 ft

1 
dt
8
4 sec
The ladder is
moving down the
wall at ¾ ft/sec
when it is 6 ft. from
the wall.
Water is draining from a cylindrical
tank at 3 liters/second. How fast is
the surface dropping?
3
cm
dV
L
 3000
 3
sec
dt
sec
dh
Find
dt
(We need a formula to
relate V and h. )
V   r 2h
dV
2 dh
r
dt
dt
cm3
2 dh
3000
r
sec
dt
(r is a constant.)
cm3
3000
dh
sec

dt
 r2
Hot Air Balloon Problem:
Given:
 d
rad

 0.14
min
4 dt
How fast is the balloon rising?
dh
Find
dt
h
tan  
500
d
1 dh
2
sec 

dt 500 dt

1 dh

 sec   0.14  
4
500 dt

h

500ft
2

Hot Air Balloon Problem:
Given:
 d
rad

 0.14
min
4 dt
2
How fast is the balloon rising?
dh
Find
dt
h
tan  
500
d
1 dh
2
sec 

dt 500 dt

1 dh

 sec   0.14  
4
500 dt

2
1

4
1
sec
 

4
h

500ft
 2
dh
2  0.14   500 
dt
2
ft
dh
140

min dt

Air is being pumped into a balloon at a rate of 100 cubic cm /sec. How fast
is the radius of the balloon increasing when the diameter is 50 cm?
4 3
V  r
3
dV
cm3
 100
dt
sec
We want dr/dt when d=50
or r = 25
dV
2 dr
 4r
dt
dt
1 dV dr

2
4r dt
dt
1
dr
(100) 
2
4 (25)
dt
1 cm
dr

sec
25
dt

Batman and Scooby Doo are having lunch
together when they both simultaneously
receive a call. Batman heads off to
Gotham city traveling east at 40 miles per
hour. Scooby hops in the mystery machine
and heads north at 30 miles an hour. How
fast is the distance between them
changing 6 minutes later?
The batmobile travels east at 40 mi/hr.
The mystery machine travels north at 30 mi/hr.
How fast is the distance between the
vehicles changing 6 minutes later?
r t  d
1
40   4
10
1
30   3
10
32  42  z 2
9  16  z 2
25  z 2
5 z
B
z 5
y3
A
x4
Batman travels east at 40 mi/hr.
Scooby travels north at 30 mi/hr.
How fast is the distance between the
vehicles changing 6 minutes later?
r  t x d y  z
2
2
2
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
B
z 5
y3
dy
 30
dt
A
x  4 dx  40
dt
miles
50
hour