AP CALCULUS
SUPPLEMENTARY RELATED RATES
PROBLEMS
:
1)
If a snowball melts so that its surface area decreases at a rate of
2
2
min
cm
, find the rate at which the diameter
changes when the snowball has radius
5
cm
.
2)
Boyle’s Law
states that when a sample of gas is compressed at a constant temperature, the pressure
P
and the
volume
V
satisfy the equation
PV
C
=
where
C
is a constant.
Suppose that at a certain instant, the volume is
800
3
cm
and is increasing at a rate of
10
3
s
cm
while the pressure is
100
kPa
.
At what rate is the pressure
changing at this instant?
3)
When air expands
adiabatically
(without gaining or losing heat), its pressure
P
and volume
V
are related by the
equation
PV
C
1 4
.
=
where
C
is a constant.
Suppose that at a certain instant, the volume is
500
3
cm
and the
pressure is
90
kPa
and is decreasing at a rate of
5
min
kPa
.
At what rate is the volume changing at this instant?
4)
A rocket is launched vertically and is tracked by an observer 3
km
away.
If the observer’s angle of elevation to the
rocket is increasing at
2
o
per second, what is the rocket’s velocity at the instant the angle of elevation is
π
6
radians?
5)
Water is being collected from a block of ice with a square base.
The ice is melting in such a way that the edge of
the base of the block is decreasing at
4
cm
hr
, while the block’s height is decreasing at
3
cm
hr
.
At what rate is
the water flowing into the pan when the base of the block has edge length
20
cm
and the block’s volume is
6000
3
cm
?
6)
Water is being poured into a rectangular prism (see diagram to the right) at a rate of
100
3
m
hr
.
What is the rate at which the water level is rising when the height of the water
level is
15
m
?
7)
Water is being pumped from a square pond with side length
40
m
into a circular pond with a
radius of
10
m
.
If the water level in the square pond is going down at a rate of
15
cm
min
,
then how fast is the water level in the circular pond changing?
8)
A hemispherical water tank with radius 15
m
, depth
h
meters and volume
V
cubic meters is defined by the equation
(
)
(
)
V
h
h
=
−
π
3
45
2
.
If the water tank is full and a plug at the bottom is pulled, then
Torricelli’s Law
states that
dV
dt
k
h
= −
, where
k
is a positive constant.
Find a formula for
dh
dt
in terms of
h
only.
9)
A point moves along the curve
y
x
2
3
=
in such a way that its distance from the origin increases at the constant rate
of
2
units per second.
Find
dx
dt
at the point
(
)
2 2
2
,
.
10)
When two electrical resistors,
R
1
ohms
(
)
Ω
and
R
2
ohms
(
)
Ω
are connected parallel in a circuit, then the
effective
resistance
is
1
1
1
1
2
R
R
R
eff
=
+
.
Suppose that
R
1
is increasing at a rate of
10
Ω
s
and
R
2
is decreasing at a rate of
2
Ω
s
.
Find the rate at which the
effective resistance
is changing when
R
1
100
=
Ω
and
R
2
400
=
Ω
.
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 Spring '09
 TANNY
 Calculus, Trigraph, Orders of magnitude, dt dt

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