Section 4.6 – Related Rates
5.5
I can use implicit differentiation to solve related rate word problems.
Day 1:
Find the slope the following at x = 1: 2ln(2 ) 5xy x
2If , find wA r r 2d
hen andd
t
A
dt
r3
d
2A 2 A 4
2A r
2dA
t dtdr
dr
2dA
dt2 3
dA
dt12
A 2 rh r 2, hdh
2dt
If , fid
nd whA
16d
en , and .d
4r
t
d
t
A 2 2 4 A 16
A 2 r h
dr
d
dA
d
dh2
t tr
th 2
d
dr
dt1 2 2 26 4 2
dr1
dt
If , find whenr h 4
r 2, h ,dh
dtand
d12
3 h
r 1
d.
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 2If , find when , , .A R h A 10, R 8dR 1
dt
dh 1
dt 3
A
2
d
dt
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
2 10 21
2
dA
d8
1
t 32 6
dA 3
dt 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
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
10
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.
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
d
d
t
x
d2x 2y 2
t dtR
dR
2 24 2 10 2 26dR
01
d2 t
dR 6
dt 13
The rope is being pulled in at a rate of 6/13
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
2A 32 1024
2dA
t dtdr
dr
2d
dt2
A43
dA256
dt
The area is increasing at a rate of 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?
5dr
10dt
dV
dt
34V r
3
34 500V 5 V
3 3
2d4 r
dt t
V
d
dr
2dV1004 5 0
t10
d
Air must be removed at a rate of 1000
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
2A 100 10000
2dA
t dtdr
dr
2d
dt0 4
A10
dA800
dt
The area of the spill is increasing at a rate of 800
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
15
dV
dt
2hdV
dtdt
dh
2V
dt4
d15
dV900
dt
The sand is pouring from the chute at a rate of 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 4
3
2
rate
of 30 cu ft per hour.
Water is flowing into a spherical tank with at the constant
When the water is h feet deep, the volume of water
hin the tank is given by
6 f
V 18 h . What is the3
oot radius
rate at which the depth
of the water in the ta
when the watenk is increasing 2 ft dr is 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
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
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
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
dx2 E
dt