myers (nmm698) – HW08 – Hamrick – (54868)
1
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001
10.0 points
The radius of a circle is increasing at a
constant rate of 8 ft/sec.
Express the rate at which the area of the
circle is changing in terms of the circumfer
ence,
C
of the circle.
1.
rate = 16
C
sq. ft./sec
2.
rate = 8
C
sq. ft./sec
correct
3.
rate = 4
πC
sq. ft./sec
4.
rate = 4
C
sq. ft./sec
5.
rate = 16
πC
sq. ft./sec
6.
rate = 8
πC
sq. ft./sec
Explanation:
The area and circumference of a circle of
radius
r
are given by
A
=
πr
2
.
C
= 2
πr
respectively. Thus
dA
dt
= 2
πr
dr
dt
=
C
dr
dt
.
When
dr/dt
= 8 ft/sec, therefore,
dA
dt
= 8
C
sq. ft./sec
.
002
10.0 points
A point is moving on the graph of
4
x
3
+ 3
y
3
=
xy.
When the point is at
P
=
parenleftBig
1
7
,
1
7
parenrightBig
,
its
y
coordinate is increasing at a speed of 4
units per second.
What is the speed of the
x
coordinate at
that time and in which direction is the
x

coordinate moving?
1.
speed = 2 units/sec
,
decreasing
x
2.
speed =
7
5
units/sec
,
increasing
x
3.
speed = 2 units/sec
,
increasing
x
4.
speed =
9
5
units/sec
,
decreasing
x
5.
speed
=
8
5
units/sec
,
decreasing
x
correct
6.
speed =
9
5
units/sec
,
increasing
x
7.
speed =
7
5
units/sec
,
decreasing
x
8.
speed =
8
5
units/sec
,
increasing
x
Explanation:
Differentiating
4
x
3
+ 3
y
3
=
xy
implicitly with respect to
t
we see that
12
x
2
dx
dt
+ 9
y
2
dy
dt
=
y
dx
dt
+
x
dy
dt
.
Thus
dx
dt
=
parenleftBig
x

9
y
2
12
x
2

y
parenrightBig
dy
dt
.
Now at
P
,
x

9
y
2
=

2
49
,
while
12
x
2

y
=
5
49
.
Hence, at
P
,
dx
dt
=

2
5
dy
dt
.
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myers (nmm698) – HW08 – Hamrick – (54868)
2
When the
y
coordinate at
P
is increasing at a
rate of 4 units per second, therefore,
dx
dt
=

8
5
.
Consequently, the
x
coordinate is moving at
speed
=
8
5
units/sec
,
and the negative sign indicates that it is mov
ing in the direction of decreasing
x
.
003
10.0 points
If a snowball melts so that its surface area
decreases at a rate of 1 cm
2
/
min, find the
rate at which the diameter decreases when
the diameter is 14 cm.
1.
1
42
π
cm
/
min
2.
1
56
π
cm
/
min
3.
1
7
π
cm
/
min
4.
1
28
π
cm
/
min
correct
5.
1
14
π
cm
/
min
Explanation:
If the radius is
r
and the diameter is
x
= 2
r
then
r
=
1
2
x
and
S
= 4
πr
2
= 4
π
parenleftbigg
1
2
x
parenrightbigg
2
=
πx
2
dS
dt
=
dS
dx
dx
dt
= 2
πx
dx
dt
dS
dt
=

1
2
πx
dx
dt
=

1
dx
dt
=

1
2
πx
When
x
= 14
,
dx
dt
=

1
28
π
, so the rate of
decrease is
1
28
π
cm
/
min.
004
10.0 points
A street light is on top of a 12 foot pole.
Joe, who is 3 feet tall, walks away from the
pole at a rate of 6 feet per second.
At what
speed is the tip of Joe’s shadow moving from
the base of the pole when he is 14 feet from
the pole?
1.
tip speed
= 7 ft/sec
2.
tip speed
= 9 ft/sec
3.
tip speed
= 6 ft/sec
4.
tip speed
= 8 ft/sec
correct
5.
tip speed
= 10 ft/sec
Explanation:
If
x
denotes the distance of the tip of Joe’s
shadow from the pole and
y
denotes the dis
tance of the person from the pole, then the
shadow and the lightpole are related in the
following diagram
(0
,
12)
(14
,
3)
y
x
By similar triangles,
3
x

y
=
12
x
,
so (12

3)
x
= 12
y
.
Thus, after implicit
differentiation with respect to
t
,
(12

3)
dx
dt
= 12
dy
dt
.
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 Fall '08
 schultz
 Trigraph, Imperial units, United States customary units, Inch

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