related rates section 4.6. first, a review problem: consider a sphere of radius 10cm. if the radius...
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Related Rates
Section 4.6
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.
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
Take derivative with respect to TIME.
(Possible if the sphere is a soap bubble or a balloon.)
At what rate is the the sphere?
Water is draining from a cylindrical tank of radius 3 at 3000 cm3/second. How fast is the surface dropping?
dV
dt
3cm3000
sec
Finddh
dt2 9V r h h
9dV dh
dt dt
3cm3000 9
sec
dh
dt
31000 cmsec3
dh
dt
(We need a formula to relate V and h. )
The water level is dropping at -1000/3π cm/sec.
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
rad0.14
min
d
dt
How fast is the balloon rising?
Finddh
dt
tan500
h
2 1sec
500
d dh
dt dt
2 1
2 0.14500
dh
dt
h
500ft
x
y
30dy
dt
40dx
dt
B
A
z
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.
p
A 14 foot ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2 ft/s, how fast will the end be moving away from the wall when the top is 6 ft above the ground?
14
x
y L
dx
dt
2 2 2x y L dy
2dt
dL0
dt
2 2 2x 6 14 x 4 10
dy
d
d
t
x
d2x 2y 2
t dtL
dL
2 4 10 2 6 2 1 0dx
d2 4
t
dx 3
dt 10
The ladder is moving away at a rate of 3
10How fast is the area changing?
A 14 foot ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2 ft/s, let’s now find how fast the area of the triangle is changing when the top is 6 ft above the ground?
14
x
y L
dx 3
dt 10
dy2
dt dL
0dt
A man 6 ft tall is walking at a rate of 2 ft/s toward a street light 16 ft tall. At what rate is the size of his shadow changing?
616
x y
6 x
16 x y
dy2
dt
dx
dt
6x 6y 16x
10x 6y 0 dx
dt10 6
t0
dy
d
dx
dt10 6 2 0
dx 6
dt 5
The size of his shadow is reducing at a rate of 6/5 ft/s.
A boat whose deck is 10 ft below the level of a dock, is being drawn in by means of a rope attached to a pulley on the dock. When the boat is 24 ft away and approaching the dock at ½ ft/sec, how fast is the rope being pulled in?
x
y R
dx 1
dt 2
dy0
dt dR
dt
2 2 2
2 2 2
x y R
24 10 R
R 26dy
d
d
t
x
d2x 2y 2
t dtR
dR
2 24 2 10 2 21
02
6dR
dt
dR 6
dt 13
The rope is being pulled in at a rate of 6/13 ft/sec.
A pebble is dropped into a still pool and sends out a circular ripple whose radius increases at a constant rate of 4 ft/s. How fast is the area of the region enclosed by the ripple increasing at the end of 8 seconds.
dr4
dt
dA
dt
2A r
At t = 8, r = (8)(4) = 32
2dA
t dtdr
dr
2d
dt2
A43
dA256
dt
The area is increasing at a rate of ft2/sec. 256
A spherical container is deflated such that its radius decreases at a constant rate of 10 cm/min. At what rate must air be removed when the radius is 5 cm?
dr10
dt
dV
dt
34V r
3
2d4 r
dt t
V
d
dr
21
dV1000
dt4 5 0
Air must be removed at a rate of ft3/min. 1000
Sand pours into a conical pile whose height is always one half its diameter. If the height increases at a constant rate of 4 ft/min, at what rate is sand pouring from the chute when the pile is 15 ft high?
21V r h
3
1h d
2
1h 2r
2
h r
31V h
3 dh
4dt
dV
dt
2hdV
dtdt
dh
2V
dt4
d15
dV900
dt
The sand is pouring from the chute at a rate of ft3/min. 900
Liquid is pouring through a cone shaped filter at a rate of 3 cubic inches per minute. Assume that the height of the cone is 12 inches and the radius of the base of the cone is 3 inches. How rapidly is the depth of the liquid in the filter decreasing when the level is 6 inches deep?
dV3
dt
12
3
h
r
21V
3hr
r
3 2
h
1
r h1
4
2
V h1
3
1
4h
3V h1
48
23
48
dhh
d
dt
V
dt
236
48
h
dt3
d
4 dh
3 dt
The depth of the liquid is decreasing at a rate of in/sec. 4
3
If and x is decreasing at the rate of 3 units per second,the rate at which y is changing when y = 2 is nearest to:
2xy 20
a. –0.6 u/s b. –0.2 u/s c. 0.2 u/s d. 0.6 u/s e. 1.0 u/s
2xy 20
2x 2 20
x 5
2y 2ydy
d
dx
dt0
tx
2 dy
dt2 2 2 53 0
dy
dt20 12
dy
dt
3
5
2
dy8
d
A particle moves along a curve x y 2 at time t 0.
If when , what is the value of at that timdx
dtx 1
te?
2x y 2
2y 2-1 y 2
2dyd
d
x
d t2 0x y x
t
2dx
dt1 2 8 02 1
dx
2dt
dx
dt4 8