4.6: Related Rates
Olympic National Park, Washington
Photo by Vickie Kelly, 2007
Greg Kelly, Hanford High School, Richland, Washington
4.6: Related Rates
Olympic National Park, Washington
Photo by Vickie Kelly, 2007
Greg Kelly, Hanford High School, Richland, Washington
4.6: Related Rates
Olympic National Park, Washington
Photo by Vickie Kelly, 2007
Greg Kelly, Hanford High School, Richland, Washington
First, a review problem:
Consider a sphere of radius 10cm.
If the radius changes 0.1cm (a very small amount) how
much does the volume change?
4
V   r3
3
dV  4 r 2 dr
dV  4 10cm   0.1cm
2
dV  40 cm3
The volume would change by approximately 40 cm 3 .

Now, suppose that the radius is changing at an
instantaneous rate of 0.1 cm/sec.
(Possible if the sphere is a soap bubble or a balloon.)
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 .
Note: This is an exact answer, not an approximation like
we got with the differential problems.

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

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.

Hot Air Balloon Problem:
Given:

 d
4
rad
 0.14
dt
min
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
4
rad
 0.14
dt
min
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

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  d
1
40   4
10
1
30   3
10
32  42  z 2
B
z 5
y3
A
x4
9  16  z 2
25  z 2
5 z

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
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
