Section 4.6 – Related Rates

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5.5

2 dA

dt1. If , find when andA r 32

dtr

dr

2A r

2dA

t dtdr

dr

2dA

dt2 3

dA

dt12

2. If , find when , and .ddr

dt

A16

d

dh2A

dt2 rh r 2, 4

th

A 2 r h

dr

d

dA

d

dh2

t tr

th 2

d

dr

dt1 2 2 26 4 2

dr1

dt

3. If , find wher h 4

r 2, h 12n , andh

dd

3 h

r 1

dtd.

t 2

11r 1 4h

3

214h

dh

d

dr

d3 tt

2

d1 4

3 t2

1

d1

h

2

dh6

dt

2 2 2A R h A 10, R4. I 8f , fidRdA

dt

1

dtnd when

2, ,

dt 3.

dh 1

2 2 2 2 2 2A R h 10 8 h h 6

2 2 2A R h

dR

d

d

t

A

d2A 2R 2

t dth

dh

10 81

2

dA

d

1

t 36

dA 3

dt 5

5. 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?

146

x

y L

dx

dt

2 2 2x y L dy

2dt

dL0

dt

2 2 2x 6 14 x 4 10

4 10dy

dt

dx

dx y L

t

dL

dt

dx

dt4 10 6 1 02 4

dx 3

dt 10

The ladder is moving away at a rate of 3

10

7. 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

3x 3y 8x

5x 3y 0

5 3dd yx

d t0

t d

dx

d5 2

t3 0

dx 6

dt 5

The size of his shadow is reducing at a rate of 6/5.

3 x

8 x y

8. 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?

-10

24x

y R

dx 1

dt 2

dy0

dt

dR

dt

2 2 2

2 2 2

x y R

24 10 R

R 26

26

dy

dt

dx

dx y R

t

dR

dt

d024 10 26

1 R

2 dt

dR 6

dt 13

The rope is being pulled in at a rate of 6/13

9. 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 256

10. 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?

5dr

10dt

dV

dt

34V r

3

2d4 r

dt t

V

d

dr

2dV1004 5 0

t10

d

Air must be removed at a rate of 1000

11. A ruptured pipe of an offshore oil platform spills oil in a circular pattern whose radius increases at a constant rate of 4 ft/sec. How fast is the area of the spill increasing when the radius of the spill is 100 ft?

dr4

dt

100

dA

dt

2A r

2dA

t dtdr

dr

2d

dt0 4

A10

dA800

dt

The area of the spill is increasing at a rate of 800

12. 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

15

dV

dt

2hdV

dtdt

dh

2V

dt4

d15

dV900

dt

The sand is pouring from the chute at a rate of 900

13. 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 4

3

14. A trough is 15 feet long and 4 feet across the top. Its ends are isosceles triangles with height 3 ft. Water runs into the trough at the rate of 2.5 cubic feet/min. How fast is the water level rising when it is 2 feet deep?

15

4

3 y

xL

1V x L

2y dL

0dt

3y

V y15

x2

2

x

2x2y

2

3 3

2x

y2

4x y

3

4y

15y

2 3V

20dy

ydV

d tt d dy

2205

t2 d

1 dy

16 dt

The water level is rising at a rate of 1/16.

2

rate of 30 cu ft per ho

15. Water is flowing into a spherical tank with at the constant

When the water is h feet deep, the volume

hof water in the tank is given by V

6 foot radi

1

ur

8 h . What is

.

3

us

rate at which

the depth of the water in the

the

when the watank is incr ter iseasing 2 ft d eep?

6

dV30

dt

dh

dt 2

32 h

V 6 h3

2dh dhh h

dt12

dV

dtdt

dh dh2 4

dt30

t2

d1

dh 3C

dt 2

16. If and x is decreasing at the rate of 3 units persecond, 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

17. When a wholesale producer market has x crates of lettuce available on a given day, it charges p dollars per crate as determined by the supply equation If the daily supply is decreasing at the rate of 8 crates per day, at what rate is the price changing when the supply is 100 crates?

px 20p 6x 40 0

px 20p 6x 40 0

p 20p 6100 10 400 0 p 7

dp dp

dt dt

dx dx

dt dx p 2 6

t0 0

8d

100 7 2p dp

dt dt0 6 8 0

dp0.1 B

dt

2

dy8

dt

18. A particle moves along a curve x y 2 at time t 0.

If when , what is the vadx

dtlue of at that t me?x 1 i

2x y 2

2y 2-1 y 2

2dyd

d

x

d t2 0x y x

t

2dx

dt1 2 8 02 1

dx2 E

dt