AP Calculus
Lesson- Related Rates
Name:____________________________________
Date:_____________________________________
Objectives:
Identify a mathematical relationship between quantities that are each changing.
Use one or more rates to determine another rate.
Process:
Step 1: Draw a diagram.
Step 2: Determine which quantities and rates are given, and which to be found.
Step 3: Identify the primary function to use. (Often this is a formula from geometry.)
Step 4: Differentiate with respect to the independent variable
Step 5: Write a related rates equation
Step 6: Substitute known quantities and solve for desired rate.
[NOTE: Do not substitute known quantities before this last step!]
Type 1: Explicit Function of One Variable
Examples
1.
Air is being blown into a sphere at the rate of 6 cubic inches per minute. How fast is the radius
changing when the radius of the sphere is 2 inches?
2.
What we know:
v
dv
 6 in 3 / min
dt
r  2 in
dv
dr
 4r 2
dt
dt
dr
6  4 (2) 2
dt
dr
6
1


in / min
dt 16 4
4 3
r
3
The edge of a cube is increasing at a rate of 2 inches per minute. At the instant the edge is 3 inches, how
fast is the volume increasing?
What we know:
dv
 2 in 3 / min
dt
s  3 in
v  s3
dv
ds
 3s 2
dt
dt
ds
2  3(3) 2
dt
ds
2 3

in / min
dt 27
3.
A point moves along the curve y   x  3 such that its x-coordinate is increasing at 4 units per second.
(a)
At the moment x = 1, how fast is the y-coordinate changing? Interpret your answer based on the
shape of the graph and the location of the point.
2
(b)
What we know:
y   x  3
dx
 4 units / sec
dt
x 1
dy
dx
 2( x  3)
dt
dt
dy
 2(1  3)  4  16 units / sec
dt
2
At the moment x = 1, how fast is the point’s distance from the origin changing?

What we know:
D  ( x  0) 2  ( x  3) 2  0
dx
 4 units / sec
dt
x 1
D  ( x) 2  ( x  3) 2


2

2
D  x 2  ( x  3) 4


1/ 2
D  x 2  ( x  3) 4
1 / 2
dD 1 2
dx
 x  ( x  3) 4
 2 x  4( x  3) 3 
dt 2
dt

1
/
2
dD 1 2
 (1)  (1  3) 4
 2(1)  4(1  3) 3  4
dt 2
dD 1
1 / 2
1 / 2
 17    30  4  6017   14.552 units / min
dt 2








Type 2 - Implicit Function of One Variable
4.
A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at
a rate of 1 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is
6 ft from the wall?
What we know:
dx
 1 ft / sec
dt
r  10 ft
x  6 ft
x2  y2  r 2
dx
dy
dr
2x  2 y
 2r
dt
dt
dt
dy 

(2  6  1)   2  8    0
dt 

dy
16
 12
dt
dy  12  3


ft / sec
dt
16
4
Type 3: Functions of Two Variables—2 rates given
5.
The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at
a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the
area is 100 cm2?
What we know:
1
bh
2
1
100  b  10
2
b  20 cm
1
A  bh
2
dA
1
dh
db 
 b 
 h

dt
2
dt
dt 
1
db 
2   20  1  10 

2
dt 
db
 1.6 cm / sec
dt
A
dh
 1 cm / min
dt
dA
 2 cm 2 / min
dt
h  10 cm
A  100 cm
Type 4: Functions of Two Variables—1 rate given--secondary equation needed
6.
A water tank has the shape of an inverted circular cone with base radius 2m and height 4 m. If water is
being pumped into the tank at a rate of 2m3/min, find the rate at which the water level is rising when the
water is 3 m deep.
What we know:
rbase  2 m
htotal  4 m
dv
 2 m 3 / min
dt
h3m
1 3
h
12
dv 1
dh
  h2
dt 4
dt
1
2 dh
2   3
4
dt
r 2

h 4
h
r
2
1
v   r 2h
3
v
2
1 h
v   h
3 2
dh 8

m / min
dt 9
Mixed Problem Set- Related Rates
1.
A conical tank is being filled with water. The tank has height 4 ft and radius 3 ft. If water is being
pumped in at a constant rate of 2 cubic inches per minute, find the rate at which the height of the cone
changes when the height is 26 inches. Note the difference in units.
What we know:
rbase  3 in
htotal  4 in
dv
 2 in 3 / min
dt
h  26 in
2.
v
2
1  3h 
v   h
3  4 
dh
2

in / min
dt 1521
A searchlight is positioned 10 meters from a sidewalk. A person is walking along the sidewalk at a
constant speed of 2 meters per second. The searchlight rotates so that it shines on the person. Find the
rate at which the searchlight rotates when the person is 25 meters from the searchlight.
What we have:
y  10 m
dx
 2 m / sec
dt
r  25 m
3.
3 3
h
4
dv 9
dh
  h2
dt 4
dt
9
2 dh
2   26 
4
dt
r 3

h 4
3h
r
4
1
v   r 2h
3
x2  y2  r 2
x 2  (10) 2  (25) 2
x  525
25 5
sec  

10 2
x
y
tan  
d
sec 2 

dt


y
dy
dx
 x
dt
dt
2
y
 5  d 10  2  525  0

 
10 2
 2  dt
 25  d 1

 
 4  dt 5
d
1

rad / sec
dt 25
2
A person 5 feet tall is walking toward an18 foot pole. A light is positioned at the top of the pole. Find
the rate at which the length of the person’s shadow is changing when the person is 30 feet from the pole
and walking at a constant speed of 6 feet per second.
What we have:
18 r  x

5
r
18 r  30

5
r
18 r  x

5
x
18
x
1
5
r
dx
dr
x
dt
dt
0  0
r2
18 r  30

5
r
18r  5r  150
13r  150
r  150 13
r
0r
dx
dr
x
dt
dt
0  (150 / 13)(6)  30
dr
 30 / 13 ft / sec
dt
dr
dt
4.
The length of a rectangle increases by 3 feet per minute while the width decreases by 2 feet per minute.
When the length is 15 feet and the width is 40 feet, what is the rate at which the following changes:
a. area:
A  LW
b. perimeter:
P  2 L  2W
dA
dW
dL
L
W
dt
dt
dt
dP
dL
dW
2
2
dt
dt
dt
dA
 15  2  40  3  90 ft 2 / min
dt
dP
 2  3  2  2  2 ft / min
dt
c. diagonal

5.
6.





1 / 2  dL
dr 1 2
dW  1
2
2 1 / 2
2  3  2  2
 L W 2
 2
2
  (15)  (40)
dt 2
dt  2
 dt
1

ft / min
1825
1
C 2 h where C is the circumference of the tree in meters at
The volume of a tree is given by V 
12
ground level and h is the height of the tree in meters. Both C and h are functions of time t in years.
r  L2  W 2
1/ 2
dV
. What does it represent in practical terms?
dt
dV
1  2 dh
dC 
dV
C  dh
dC 

 h  2C

 2h
C

C

dt 12 
dt
dt 
dt 12  dt
dt 
 in circumference over time
a.
Find a formula for
b.
Suppose the circumference grows at a rate of 0.2 meters/year and the height grows at a rate of 4
meters/year. How fast is the volume of the tree growing when the circumference is 5 meters and
the height is 22 meters?
dV
5
dV
5
dV 12 3
5  4  2  22  0.2
28.8



m / year
dt 12
dt 12
dt

a.
When the radius of a spherical balloon is 10 cm, how fast is the volume of the balloon changing
with respect to change in its radius?
4
dv
dr
dv
dr
dr
v  r 3
 4r 2
 4 (10) 2
 400
3
dt
dt
dt
dt
dt
b.
If the radius of the balloon is increasing by 0.5 cm/sec, at what rate is the air being blown into
the balloon when the radius is 6 cm?
4
dv
dr
dv
v  r 3
 4r 2
 4 (6) 2 (0.5)  72 cm 3 / sec
3
dt
dt
dt
7.
When hyperventilating, a person breathes in and out very rapidly. A spirogram is a machine that
draws a graph of the volume of air in a person’s lungs as a function of time. During hyperventilation, the
person’s spirogram trace might be represented by V  3  0.05 cos200t  where V is the volume of air
in liters in the lungs at time t minutes.
a.
Sketch a graph of one period of this function.
b.
What is the rate of flow of air in liters/minute? Sketch a graph of this function.
dV
V  3  0.05 cos200t 
 0.05 sin( 200t )  200  10 sin( 200t )
dt
c.
Mark the following on each of the graphs above.
i) the interval when the person is breathing in
(0.005, 0.01)
ii) the interval when the person is breathing out
(0, 0.005)
iii) the time when the rate of flow of air is a maximum when the person is breathing in
0.0075 min