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RELATED RATES PROBLEMS. - PowerPoint PPT Presentation
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RELATED RATES PROBLEMS
If a particle is moving along a straight line according to the equation of motion , since the velocity may be interpreted as a rate of change of distance with respect to time, thus we have shown that the velocity of the particle at time “t” is the derivative of “s” with respect to “t”.
)t(fs
There are many problems in which we are concerned with the rate of change of two or more related variables with respect to time, in which it is not necessary to express each of these variables directly as function of time. For example, we are given an equation involving the variables x and y, and that both x and y are functions of the third variable t, where t denotes time.
Since the rate of change of x and y with respect to t is given by and , respectively, we differentiate both sides of the given equation with respect to t by applying the chain rule.
When two or more variables, all functions of t, are related by an equation, the relation between their rates of change may be obtained by differentiating the equation with respect to t.
dx
dt
dy
dt
A Strategy for Solving Related Rates Problems (p. 205)
Example 1 A 17 ft ladder is leaning against a wall. If the bottom of the ladder is pulled along the ground away from the wall at the constant rate of 5 ft/sec, how fast will the top of the ladder be moving down the wall when it is 8 ft above the ground?
wallthe fromaway
ground the along pulled is ladder the of bottom the sincesec time t
instantany at ground the from ladder the of top the of ft distance y
instantany at wallthe from ladder the of bottom the of ft distance x Let
sec
ft5
dt
dx
x
17 ft.?
dt
dy
ft8y
y
Note: • Values which changes as time changes are denoted by variable.• The rate is positive if the variable increases as time increases and is negative if the variable decreases as time increases.
Equation Working 17yx 222
ydtdx
x
y2dtdx
x2
dtdy
0dtdy
y2dtdx
x2
secft
375.98
515dtdy
15817x8y when 22
Example 2 A balloon leaving the ground 60 feet from an observer, rises vertically at the rate 10 ft/sec . How fast is the balloon receding from the observer after 8 seconds?
ground the from rise to startsballoon the sincesec time t
instantany at observer the from balloon the of ft distance L
instantany at ground the from balloon the of ft height h Let
Viewer60 feet
h L
sec
ft10
dt
dh
?dt
dL
sec8t
Equation Working 3600hL
60hL :figure the In2
222
3600hdtdh
h
dtdL
3600h2dtdh
h2
dtdL
2
2
.ft 808sec secft
10h
8sect and secft
10dtdh
,Since
secft
8 100800
dtdL
000,10800
36006400800
dtdL
3600801080
dtdL
2
Example 3 As a man walks across a bridge at the rate of 5 ft/sec , a boat passes directly beneath him at 10 ft/sec. If the bridge is 30 feet above the water, how fast are the man and the boat separating 3 seconds later?
secft
5
secft
10
instantany at boat the and
man the between ft distance s
bridge the cross to
startsman the sec time t Let
sec 3 t when dtds
:Find
Equation Working 125t900S
125t900S
125tL but L900 S
L30S
2
22
22
222
secft
8.33 or secft
325
453125
dtdS
3125900
3125dtds
sec3t when
2
S
L5t
30’
30’ 10t
secft
10
secft
5
2
22
222
t125 L
t10t5 L
t10t5L
2
2
t125900t125
dtdS
t1259002t2125
dtdS
:t time wrt WE theof sides both ateDifferenti
R
secft
4dtdR
x
20ft
Example 4 A man on a wharf of 20 feet above the water pulls in a rope, to which a boat is attached, at the rate of 4 ft/sec. At what rate is the boat approaching the wharf when there is 25 feet of rope out?
instantany at out rope the of ft length R
instantany at wharfthe from boat the of ft distance x
wharfthe approach to startsboat the sincesec time t Let
)Equation Working( 400R x
400Rx
20xR
ft25R when dtdx
Find
2
22
222
400RdtdR
R
400R2dtdR
R2
dtdx
400Rx
22
2
secft
320
dtdx
15425
40025425
dtdx
secft
4dtdR
and
ft25R When
2
Example 5 Water is flowing into a conical reservoir 20 feet deep and 10 feet across the top, at the rate of 15 ft3/min . Find how fast the surface is rising when the water is 8 feet deep?
5 feet20
feet
h
r
10 feet
min
ft15
dt
dV 3
instantany at waterthe of (ft) heighth
instantany at surface waterthe of (ft) radius r
reservoir the oint flows water the cesin min time t Let
h r 31
Bh31 V
deep ft. 8 is water the when dtdh
Find
2h
41
rhr
205
proportion and ratio By
Equation Working h48
V hh41
31
V ,Thus 3
2
min
ft1.194 or
minft
415
81516
hdtdV
16
dtdh
dtdh
h16dt
dhh3
48dtdV
22ft8h
22
Example 6 Water is flowing into a vertical tank at the rate of 24 ft3/min . If the radius of the tank is 4 feet, how fast is the surface rising?
h
4 feet
min
ft24
dt
dV 3
instantany at waterhet of ft volume V
instantany at waterthe of ft heighth
tank the into flows waterthe sincemin time t Let
3
ft. 4 is tank theof radius the when dtdh
Find
minft
23
1624
16dtdV
dtdh
dtdh
16dtdV
ft4r
Equation Working h 16h4V
ft 4r constant, is r Buthr h r V
BhV From
2
22
Example 7 A triangular trough is 10 feet long, 6 feet across the top, and 3 feet deep. If water flows in at the rate of 12 ft3/min, find how fast the surface is rising when the water is 6 inches deep?
min
ft12
3
h
6 feet
10 feet3 fe
et x
instantany at waterhet of ft volume V
instantany at end
triangular the at waterthe of ft widthhorizontal x
instantany at waterthe of ft heighth
trough the into flows waterthe sincemin time t Let
3
deep. inches 6 is water the when dtdh
Find
Equation Working 10hh2h55xhV Thus,
h2x36
hx
,proportion and ratio yB
h5x10h x21
V
BhV From
2
minft
2.1
in12ft1
in620
12h20
dtdV
dtdh
dtdh
h20dtdV
in6h
Example 8 A train, starting at noon, travels at 40 mph going north. Another train, starting from the same point at 2:00 pm travels east at 50 mph . Find how fast the two trains are separating at 3:00 pm.
80 m
iles
x2pm
B
C
DA
L
y
3pm
3pm
hr
mi40
dt
dy
hr
mi50
dt
dx
12pm
2pm
1hr. t enwh dtdL
Find
Equation Working y80xL
y80xL :figure the From22
222
22 )y80(x2dtdy
)y80(2dtdx
x2
dtdL
miles 40hr1mph40ymiles 50hr150mphx
hr 1 After
miles 80240BA
mph 40dtdy
and mph 50dtdx
Since
22 )4080()50()40)(4080(())50)(50(
dtdL
400,14500,2800,4500,2
dtdL
22 )y80(xdtdy
)y80(dtdx
x
dtdL
130300,7
900,16300,7
dtdL
hrmi
15.56dtdL
Example 9 A billboard 10 feet high is located on the edge of a building 45 feet tall. A girl 5 feet in height approaches the building at the rate of 3.4 ft/sec . How fast is the angle subtended at her eye by the billboard changing when she is 30 feet from the billboard?
x
sec.ft
43
45’
10’
5’
ft. 30x enwh dtd
Find
Equation Working 2000xx10
tan
2000xx10
x2000xx
10
tan
2
1
2
2
2
:figure the In
x40
x50
1
x40
x50
tan
x40
tan and x
50 tan ,but
tan tan1 tan tan
tan
tan tan :gsinU
22
2
2
2
2000xdtdx
)x2(x10dtdx
)10)(2000x(
2000xx10
1
1dtd
Equation Working 2000xx10
tan2
1
222
22
x1002000xdtdx
x20000,20x10
dtd
222
2
x1002000xdtdx
x10000,20
dtd
222
2
30 100 2000304.3 30 10000,20
dtd
000,500,8
4.3 000,11dtd
secrad
0044.0dtd
Example 10 A picture 40 cm high is placed on a wall with its
base 30 cm above the level of the eye of an observer. If the observer is approaching the wall at the rate of 40 cm/sec, how fast is the measure of the angle subtended at the observer’s eye by the picture changing when the observer is 1 m from the wall?
1
2
40 cm
30 cm
1
2
40 cm
30 cm
x
12
30x
cot70x
cot 11
2
2
2
2
30x
1
dtdx
301
70x
1
dtdx
701
dtd
22 x900
90030
dt/dxx4900
490070
dt/dxdtd
10000900900
3040
1000049004900
7040
dtd
100cm1mx and 40cm/secdtdx
substitute
10900
300 4
14900700
4dtd
...07782.010912
14928
dtd
.sec/.rad 078.0dtd
A statue 10ft. high is standing on a base 13ft. high. If an observer’s eye is 5ft. above the ground, how far
should he stand from the base in order that the angle between his lines of sight to the top and bottom of the statue be a maximum?
8x
cot18x
cot
8x
cot 8x
cot
18x
cot 18x
cot
figure the From
11
111
122
12
81
8x
1
1181
18x
1
1dxd
22
Example 11
1
2
1
2
10’
13’
5’x
0
8x
1
81
18x
1
181
dxd
22
2
2
2
22
2
2
2 88
8x
1
81
1818
18x
1
181
2222 x88
x1818
22 x3244x649
22 x43244x9649
5761296x5 2
12x
Therefore, the observer must be 12 ft from the base of the statue so that his line of sight from top to bottom of the statue is maximum.
720x5 2
12x144x2
1. What number exceeds its square by the maximum amount?2. The sum of two numbers is “K”. find the minimum value of the sum of their squares.3. A rectangular field of given area is to be fenced off along the
bank of a river. If no fence is needed along the river, what are the dimensions of the rectangle that will require the least amount of fencing?
4. A Norman window consists of a rectangle surmounted by a semicircle. What shape gives the most light for a given perimeter?
5. A cylindrical glass jar has a plastic top. If the plastic is half as expensive as the glass per unit area, find the most economical
proportions for the glass.6. Find the proportions of the circular cone of maximum volume inscribed in a sphere.7. A wall 8 feet high and 24.5 feet from a house. Find the shortest ladder which will reach from the ground to the house when leaning over the wall
EXERCISE A:
1. A sign 3 ft high is placed on a wall with its base 2 ft above the eye level of a woman attempting to read it. Find how far from the wall the woman should stand to get the “best view” of the sign; that is, so that the angle subtended at her eye by the sign is maximum.
2. A man on dock is pulling in at the rate of 2ft/sec a rowboat by means of a rope. The man’s hands are 20ft. above the level of the point where the rope is attached to the boat. How fast is the measure of the angle of depression of the rope changing when there are 52 ft. of rope out?
EXERCISE B:
4. A picture 5 ft high is placed on a wall with its base 7ft above the level of the eye of an observer is approaching the wall at the rate of 3ft/sec. How fast is the measure of the angle subtended at her eye by the picture changing when the observer is 10ft. from the wall?
3. Find the equations of the normal line and tangent lines to the graph of the equation at the point .
1x2secy 1
3
1,2
1
5. An airplane is flying at a speed of 300mi/hr at an altitude of 4 mi. If an observer is on the ground, find the time rate of change of the measure of the observer’s angle of elevation of the airplane when the airplane is over a point on the ground 2 mi. from the observer.