2.6 Related Rates

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?

34

3V r

24dV r dr

24 10cm 0.1cmdV

340 cmdV

The volume would change by approximately .340 cm

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.)

34

3V r

24dV dr

rdt dt

2 cm4 10cm 0.1

sec

dV

dt

3cm

40sec

dV

dt

The sphere is growing at a rate of .340 cm / 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?

L3

sec

dV

dt

3cm3000

sec

Finddh

dt2V r h

2dV dhr

dt dt (r is a constant.)

32cm

3000sec

dhrdt

3

2

cm3000

secdh

dt r

(We need a formula to relate V and h. )

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.

Ex. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When this radius is 4 ft., what rate is the total area A of the disturbed water increasing.

Given equation:

Givens:

Differentiate:

Hot Air Balloon Problem:

Given:4

rad

0.14min

d

dt

How fast is the balloon rising?

Finddh

dt

tan500

h

2 1sec

500

d dh

dt dt

2

1sec 0.14

4 500

dh

dt

h

500ft

Hot Air Balloon Problem:

Given:4

rad

0.14min

d

dt

How fast is the balloon rising?

Finddh

dt

tan500

h

2 1sec

500

d dh

dt dt

2

1sec 0.14

4 500

dh

dt

h

500ft

2

2 0.14 500dh

dt

1

12

4

sec 24

ft140

min

dh

dt

The velocity of an airplane tracked by radar

An airplane is flying at an elevation of 6 miles on a flightpath that will take it directly over a radar tracking station. Let s represent the distance (in miles)between the radarstation and the plane. If s is decreasing at a rate of 400 miles per hour when s is 10 miles, what is the velocity of the plane.

s

x

6

Given:

Find:

Equation:

Solve:To find dx/dt, wemust first find xwhen s = 10

A fish is reeled in at a rate of 1 foot per secondfrom a bridge 15 ft. above the water. At what rate is the angle between the line and the water changing when there is 25 ft. of line out?

15 ft.

x

Given:

Find:

Equation:

Solve:

x = 25 ft. h = 15 ft.

4x

3y

B

A

5z

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

40 410

130 3

10

2 2 23 4 z 29 16 z

225 z5 z

4x

3y

30dy

dt

40dx

dt

B

A

5z

Truck Problem:

How fast is the distance between the trucks changing 6 minutes later?

r t d 1

40 410

130 3

10

2 2 23 4 z 29 16 z

225 z5 z

2 2 2x y z

2 2 2dx dy dzx y zdt dt dt

4 40 3 30 5dz

dt

250 5dz

dt

50dz

dt

miles50

hour

Truck A travels east at 40 mi/hr.Truck B travels north at 30 mi/hr.