Warmup1)

2)

4.6: Related Rates

They are related (Xmas 2013)

Suppose that the radius of a sphere is changing at a rate of 0.1 cm/sec. The sphere has a radius of 10 cm to startFind the rate of change of the volume.(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

Air is being pumped into a balloon at a rate of 6 cubic feet per minute.

Find the rate of change of the radius when the radius is 2 ft.

Ex.

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.

A pebble gets dropped into a pool and the resulting ripplesare concentric circles that expand from the center.The radius of the outer ripple is increasing at a constantrate of 1 foot per second. When the radius is 3 feet, at what rate is the total area of the water changing?

All edges of a cube are expanding at a rate of 4 inches per second. How fast is the volume changing when each edge is (a) 1 inch, (b) 3 inches ?

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

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.

Police chase problem (group):

A police cruiser, approaching a right intersection from the north, is chasing a speeding car that has turned thecorner and headed east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine the distance between them is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant measurement,what is the speed of the car?

ex. 3 from our book

Water runs into a conical tank at a rate of 9ft3/min.

The tanks stands point down and has a height of 10ftand a base radius of 5ft. How fast is the water level rising when the water is 6ft deep?

Ex. 4 p. 235

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. )

31 L 1000 cm

B) At what rate is the angle changing where theladder meets the ground when the ladder is 10 feet from from the base of the wall.

The end