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.