Warmup:

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

Now back to our volume problem, 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

r

A = r 2

Suppose a circle is growing as time passes by:

r

A

= r 2

r

A

= r 2

r

A = r 2

r

= r 2A

r

= r 2A

Both the radius r and the area A are changing with time…

… so the radius r and the area A are functions of time…

r(t)

Both the radius r and the area A are changing with time…

… so the radius r and the area A are functions of time…

= r2(t) A(t)

r(t)

a) How are dA/dt and dr/dt related?

b) When the radius is 5 cm, increasing at a rate of 2 cm/sec, at what rate is area changing?

Typical related rates question:

= r2(t) A(t)

2A t r t

dA t

dt

If A and r are both functions of time, then

a) How are dA/dt and dr/dt related?

2r t dr t

dt

2A rdA

dt 2

drr

dt

Without the t’s :

dA

dt 2

drr

dt

b) When the radius is 5 cm, increasing at a rate of 2 cm/sec, at what rate is area changing?

dA

dt 5 cm2

dr

dt

b) When the radius is 5 cm, increasing at a rate of 2 cm/sec, at what rate is area changing?

dA

dt 5 cm

2 2

cm

sec

b) When the radius is 5 cm, increasing at a rate of 2 cm/sec, at what rate is area changing?

2cm20

sec

dA

dt

Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

3

3

4rV

dt

dV 4

3

If V and r are both functions of time, then

23r dr

dt

dt

dV 24dr

rdt

Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

2

1

4

dV

r dt dr

dt

Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

2

3c1

4

m1000

minr

dr

dt

Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

2

3cm1000

1

4

min10 cm

dr

dt

Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

2

3cm1000

1

4

mi100 c nm

dr

dt

Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

5 cm

2 min

dr

dt

Water is being released from a spherical water balloon at a rate of 1000 cm3/minute. When the balloon’s radius is 10 cm, how fast is the balloon’s radius changing?

The ‘-’ sign tells us that the radius is decreasing.

5 m

1 m/s

h

One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall?

5 m

1 m/s

h

One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall?

3 m

?d

dt

5

hsin

sin5

h

1

5

dh

dt cos

If h and are both functions of time, then

d

dt

We assume RADIAN measure

1 1

5s

mm/ cos

d

dt

One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall?

1 m

5 m cos s

d

dt

One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall?

5 m

1 m/s

h

3 m

1 m

5 m 3/ 5 s

d

dt

One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall?

1

3 s

d

dt

One end of a 5 m ladder is sliding down a wall at a rate of 1 m/s. At what rate is the angle of the ladder with the floor changing when the other end is 3 m from the wall?

rad

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

dhr

dt

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.

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 dz

x 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.