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.