Related Rates
Section 2.6
Read Guidelines For Solving Related Rates Problems on p. 150.
A 17 foot ladder is sliding down a wall. The base of the ladder is moving away from the wall at a rate of 2 feet per second. How fast is the top of the ladder moving down the wall when the base of the ladder is 8 feet away from the wall?
Example 1
Step 1: Draw a sketch and label known and unknown quantities. Write down what is given and what is to be determined.
s = 17 ft.y
x
Given: 2 ft/sdx
dt
Find: when 8dy
xdt
Step 2: Write an equation involving the variables whose rates of changes either are given or to be determined.
Step 3: Differentiate each side with respect to t.
2 2 217 x y
2 2 289 d d
x ydt dt
2 2 0 dx dy
x ydt dt
Step 4: Substitute all known values for the variables and their rates of change. Then solve for the required rate of change.
The top of the ladder is moving down the wall at a rate of about −1.067 ft/s when the base of the ladder is 8 ft. from the wall.
2 2 217 x y
8 so x y 15
2 8 2 2 15 0 dy
dt16
1.067 ft/s15
dy
dt
An adventurer rides down a zip-line at a speed of 80 mph. If the angle of depression of the zip-line is 75°, how fast is the zip-liner’s altitude changing?
Example 2
75°
z h
Given: 80 mphdz
dt
Find: dh
dt
75°
z h
sin 75 h
z
sin 75 z h
sin 75 dh dz
dt dt
sin 75 80 dh
dt
77.274 mphdh
dt
The adventurer’s altitude is decreasing by a rate of about 77.274 mph when the angle of depression is 75°.
A 6 foot tall man walks away from a 22 foot street light at a speed of 8 feet per second. What is the rate of change of the length of his shadow when he is 19 feet away from the light? Also, at what rate is the tip of his shadow moving?
Example 3
226
x s
Given: 8 ft/sdx
dt
a Find: when 19ds
xdt
6
22
s
x s
3
11
s
x s
3 3 11 x s s3 8x s
3
8s x
226
x s
3
11
s
x s
3 3 11 x s s3 8x s
3
8s x
3
8
ds dx
dt dt
38 3 ft/s
8
ds
dt
The length of the man’s shadow is increasing at a rate of 3 ft/s.
226
x s
Given: 8 ft/s and 3 ft/s dx ds
dt dt
b Find: when 19dy
xdt
y = x + s
dy dx ds
dt dt dt
8 3 11
The tip of his shadow is moving at rate of 11 ft/s when he is 19 ft. from the street light.
A large spherical balloon is being inflated and its volume is increasing at a rate of 3.5 cubic feet per minute. What is the rate of change of the radius when the radius is 7 feet?
Example 4
r
Given: 3.5 cu. ft. per min.dV
dt
Find: when 7 feetdr
rdt
r
34
3 V r
2 243 4
3
dV dr drr r
dt dt dt
23.5 4 7
dr
dt
2
3.5
4 7
dr
dt
0.006 ft/mindr
dt
The radius of the spherical balloon is increasing at a rate of about 0.006 ft/sec when the radius is 7 ft.
An upside-down conical tank full of water has a “base” radius of 3 meters and a height of 5 meters. The is being drained at a rate of 2 cubic meters per meter. What is the rate of change of the height of the water when the height is 4 meters?
Example 5
3
5r
h
Given: 2 cu. meters per mindV
dt
Find: when 4dh
hdt
3
5 r
h
21
3 V r h
21 3
3 5
V h h
3
5r
h
3 5
r h
3
5r h
33
25 V h
233
25
dV dhh
dt dt
92 16
25
dh
dt
3
5r
h
33
25 V h
0.111 meters/mindh
dt
The height of the water is decreasing at a rate of about 0.111 meter/min when its height is 4 meters.
Example 6
The Grand Finale!!!!
An upside-down conical tank full of water has a “base” radius of 5 feet and a height of 7 feet. The water is being drained into a cylindrical tank with radius of 5 feet and height 6 feet. The radius of the water in the conical tank is decreasing at a rate of 2 feet per minute. At what rate does the water level in the cylindrical tank rise when the water level in the conical tank is 3 feet?
5
7r
h1
5
65
h2
Given: 2 ft/mindr
dt
21Find: when 3 ft
dhh
dt
5
7r
h1
5
65
h2
2 2cone 1 cyl 2
1 and
3 V r h V r h
1
7 5
h r5
7 r
h1
1
7
5h r
2cone
1 7
3 5
V r r
3cone
7
15 V r
2
cyl 2 25 25 V h h
5
7r
h1
5
65
h2
3cone cyl 2
7 and 25
15 V r V h
cyl 225 dV dh
dt dt
2cone 73
15
dV drr
dt dt
2cone 72
5
dVr
dt214
5 r
cyl conedV dV
dt dt
5
7r
h1
5
65
h2
1
7
5h r
73
5 r
7
15r
cyl conedV dV
dt dt
214
5 r
2
2 14 1525
5 7
dh
dt2 0.514 ft/min
dh
dt
The water level in the cylindrical tank is increasing at rate of about 0.514 ft./min when the water level of the conical tank is 3 ft.