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.