Related Rates Problem Lesson 18

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 1A 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 increase and is negative if the variable decreases as time increases.

sec

ft375.9

8

515

dt

dy

15817x8y when

ydtdx

x

y2dtdx

x2

dt

dy

0dt

dyy2

dt

dxx2

Equation Working 17yx

22

222

Example 2A 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 In

2

222

3600h

dtdh

h

dt

dL

3600h2

dtdh

h2

dt

dL

2

2

.ft 808sec sec

ft10h

8sect and sec

ft10

dt

dh ,Since

sec

ft 8

100

800

dt

dL

000,10

800

36006400

800

dt

dL

360080

1080

dt

dL2

Example 3As 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

2

22

222

t125 L

t10t5 L

t10t5L

sec3 t whendt

ds :Find

2

2

2

22

22

222

t125900

t125

dt

dSt1259002

t2125

dt

dS

:t time wrtWE the of sidesboth ateDifferenti

Equation Working 125t900S

125t900S

125tL but L900 S

L30S

sec

ft8.33 or

sec

ft

3

25

45

3125

dt

dS

3125900

3125

dt

ds

sec3t when

2

S

L5t

30’

30’ 10t

secft

10

secft

5

R

secft

4dtdR

x

20ft

Example 4A 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 dt

dx Find

2

22

222

400RdtdR

R

400R2dtdR

R2

dt

dx

400Rx

22

2

sec

ft

3

20

dt

dx

15

425

40025

425

dt

dxsec

ft4

dt

dR and

ft25R When

2

Example 5Water 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 3

1 Bh

3

1 V

deep ft 8 is waterthe whendt

dh Find

2 h4

1r

h

r

20

5

proportion and ratio By

min

ft1.194 or

min

ft

4

15

8

1516

hdtdV

16

dt

dh

dt

dhh

16dt

dhh3

48dt

dV Equation Working h

48V

hh4

1

3

1V ,Thus

22ft8h

223

2

Example 6Water 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

Equation Working h 16h4V

ft 4r constant, is r But

hr h r V

BhV From

ft. 4 is tank the of radius the whendt

dh Find

2

22

min

ft

2

3

16

24

16dt

dV

dt

dh

dt

dh16

dt

dV

ft4r

Example 7A 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

Equation Working 10hh2h55xhV Thus,

h2x3

6

h

x ,proportion and ratioy B

h5x10h x2

1V

BhV From

deep. inches 6 is waterthe whendt

dh Find

2

min

ft2.1

in12

ft1in620

12

h20dt

dV

dt

dh

dt

dhh20

dt

dV

in6h

Example 8A 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

miles 80240BA

mph 40dt

dy and mph 50

dt

dx Since

1hr. t enwh dt

dL Find

Equation Working y80x L

y80xL :figure the From

22

222

22 )y80(x2dtdy

)y80(2dtdx

x2

dt

dL

miles 40hr1mph40y

miles 50hr150mphx hr 1 After

22 )4080()50(

)40)(4080(())50)(50(

dt

dL

400,14500,2

800,4500,2

dt

dL

22 )y80(xdtdy

)y80(dtdx

x

dt

dL

130

300,7

900,16

300,7

dt

dL

hr

mi15.56

dt

dL

Example 9A 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’

x40

x50

1

x40

x50

tan

x

40 tan and

x

50 tan ,but

tan tan1

tan tan tan

tan tan

:gsinU

:figure the In

ft 30 x enwh dt

d Find

ationWorkingEqu 2000x

x10tan

2000x

x10

x2000xx

10

tan

21

2

2

2

22

2

2

2

2000xdtdx

)x2(x10dtdx

)10)(2000x(

2000xx10

1

1

dt

d

Equation Working 2000x

x10tan

21

222

22

x1002000xdtdx

x20000,20x10

dt

d

222

2

x1002000xdtdx

x10000,20

dt

d

222

2

30 100 200030

4.3 30 10000,20

dt

d

000,500,8

4.3 000,11

dt

d

sec

rad0044.0

dt

d

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

EXERCISES: