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Olympic National Park, WashingtonGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007
4.6: Related Rates
Olympic National Park, WashingtonGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007
4.6: Related Rates
Olympic National Park, WashingtonGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007
4.6: 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.
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.