Section 4.6 – Related Rates
5.5
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1. If A  r 2 , find
dA
dr
when r  2 and
3
dt
dt
A  r 2
dA
dr
 2r
dt
dt
dA
 2  2  3 
dt
dA
 12
dt
dr
dA
dh
2. If A  2rh, find
when r  2, h  4,
 16 and
 2.
dt
dt
dt
A   2r  h 
dA  dr 
 dh 
  2   h    2r   
dt  dt 
 dt 
 dr 
1 6    2    4    2  2    2 
 dt 
dr
1
dt
3. If
r h4
dh
dr 1

, find
when r  2, h  12, and
 .
3
h
dt
dt 2
1
r  1  4h1
3
1 dr
2 dh
 4h
3 dt
dt
1 1
4 dh

3  2  122 dt
dh
6
dt
dA
dR 1 dh 1
4. If A  R  h , find
when A  10, R  8,
 ,
 .
dt
dt 2 dt 3
2
2
2
A 2  R2  h2  102  82  h2  h  6
A 2  R2  h2
dA
dR
dh
 2R
 2h
dt
dt
dt
dA
1
1
10    8    6 
dt
2
3
2A
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?
x 2  y 2  L2
dy
 2
dt
6 y
x 2  62  142
dL
14 dt  0
L
x
dx 4 10
dt
x  4 10
x

dx
dy
dL
y
L
dt
dt
dt

dx
4 10
  6  2   14  0 
dt
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?
6
x
3
x


16 x  y
8 xy
16
6
x
dx
dt
y
dy
 2
dt
3x  3y  8x
5x  3y  0
dx
dy
5
3
0
dt
dt
dx
5
 3  2   0
dt
dx 6

dt
5
The size of his shadow is reducing at a rate of 6/5.
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?
dy
0
dt
-10 y
dx 1

24 dt 2
x
R
26
dR
dt
x 2  y 2  R2
 24    10   R2
2
2
R  26
dx
dy
dR
x
y
R
dt
dt
dt
dR
 1
 24      10  0    26 
dt
 2
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.
A  r 2
At t = 8, r = (8)(4) = 32
dr
4
dt
dA
dt
dA
dr
 2r
dt
dt
dA
 2  32  4 
dt
dA
 256
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?
4 3
V  r
3
5
dr
 10
dt
dV
dt
dV
2 dr
 4r
dt
dt
dV
 452  10   1000
dt
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?
A  r 2
100
dr
4
dt
dA
dt
dA
dr
 2r
dt
dt
dA
 2 100  4 
dt
dA
 800
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?
1
1 2
h

d
V  r h
2
3
1
1
3
h   2r 
dh
V


h
15
4
2
3
dt
hr
dV
2 dh
 h
dt
dt
dV
dt
dV
2
  15   4 
dt
dV
 900
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?
3
1 2
V  r h
3
2
r
12
h
dV
 3
dt
1 1 
V   h h
3 4 
1
V
h3
48
dV
3
2 dh

h
dt 48
dt
3
2 dh
3 
6
48
dt
4 dh

3 dt
r
h

3 12
1
r h
4
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?
1
dL
4
V  xyL
0
2
dt
x
15
15
L
V
xy
2
15  4 
3 y
V
yy

2 3 
dV
dy
5
dy
 20 y
2
 20  2 
dt
dt
2
dt
x
1 dy
x

3
2
2
3x
4
16 dt
2
 
 2y  x  y
y
y 3
2
3
The water level is rising at a rate of 1/16.
15. Water is flowing into a spherical tank with 6 foot radius at the constant
rate of 30 cu ft per hour. When the water is h feet deep, the volume
h2
of water in the tank is given by V 
18  h . What is the rate at which
3
the depth of the water in the tank is increasing when the water is 2 ft deep?
h3
V  6h 
3
dV
dh
2 dh
 12h
 h
dt
dt
dt
dh
dh
30  12  2 
   4
dt
dt
dh 3
 C
dt 2
2
6
dh
2
dt
dV
 30
dt
16. If xy2  20 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:
a. –0.6 u/s
b. –0.2 u/s
xy2  20
x  2   20
x5
2
c. 0.2 u/s
d. 0.6 u/s e. 1.0 u/s
 dx  2  dy 
 dt  y   2y dt   x   0
 


dy 

2
 3  2   2  2    5   0
dt 

 
 
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 px  20p  6x  40  0
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
p 100   20p  6 100   40  0
p7
dp
dx
 dp 
 dx 
 dt   x    dt   p   20 dt  6 dt  0


 
dp
 dp 
 dt  100    8  7   20 dt  6  8   0


dp
 0.1  B
dt
18. A particle moves along a curve x 2 y  2 at time t  0.
dx
dy
at that time?
 8 when x  1, what is the value of
If
dt
dt
x2 y  2
2
-1
  y2y2
 dx 
 dy  2
 2 x dt   y    dt  x  0


 
 


dx 
2

2

1
2

8

1
   dt         0


dx
 2E
dt