Version 117 – K Exam 2 – Hamrick – (54868)
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have
19
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001
10.0 points
Find the rate at which the surface area of
a cube is changing with respect to its side
length
x
when
x
= 2 cm.
1.
rate = 64
π
cm
2
/
cm
2.
rate = 16
π
cm
2
/
cm
3.
rate = 48 cm
2
/
cm
4.
rate = 24 cm
2
/
cm
correct
5.
rate = 4 cm
2
/
cm
Explanation:
For a cube with side length
x
its
surface area = 6
x
2
.
Now the rate at which the surface area is
changing with respect to
x
is the derivative of
surface area with respect to
x
. Thus
rate =
d
dx
(surface area) = 12
x .
Consequently, when
x
= 2 cm.,
rate = 24 cm
2
/
cm
.
002
10.0 points
The cost function for Levi Strauss to pro
duce
x
pairs of blue jeans is
C
(
x
) = 3700 + 4
x
−
2
25
x
2
+
9
10000
x
3
.
Find the marginal cost to Levi Strauss of
producing 100 pairs of blue jeans.
1.
marginal cost = $15 per pair
correct
2.
marginal cost = $16 per pair
3.
marginal cost = $13 per pair
4.
marginal cost = $12 per pair
5.
marginal cost = $14 per pair
Explanation:
By definition,
the Marginal cost is the
derivative,
C
′
(
x
), of the cost function
C
(
x
).
Now
C
′
(
x
) = 4
−
4
25
x
+
27
10000
x
2
.
When
x
= 100, therefore,
marginal cost =
C
′
(100) = $15 per pair
.
003
10.0 points
A street light is on top of a 14 foot pole. A
person who is 4 feet tall walks away from the
pole at a rate of 5 feet per second.
At what
speed is the length of the person’s shadow
growing?
1.
speed
= 2 ft/sec
correct
2.
speed
=
19
10
ft/sec
3.
speed
=
9
5
ft/sec
4.
speed
=
17
10
ft/sec
5.
speed
=
8
5
ft/sec
Explanation:
If
x
denotes the length of the person’s
shadow and
y
denotes the distance of the
person from the pole, then shadow and the
lightpole are related in the following diagram
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Version 117 – K Exam 2 – Hamrick – (54868)
2
14
4
y
x
x
+
y
By similar triangles,
4
x
=
14
x
+
y
,
so 4
y
= (14
−
4)
x
. Thus, after implicit differ
entiation with respect to
t
,
4
dy
dt
= (14
−
4)
dx
dt
.
When
dy/dt
= 5
,
therefore, the length of the
person’s shadow is growing with
speed
= 2 ft/sec
.
004
10.0 points
A weather balloon is rising vertically at 60
meters per minute.
An observer is standing
on the ground 54 meters from the point at
which the balloon was released.
Determine
(in meters per minute) the rate at which the
distance between the feet of the observer and
the balloon is changing when the balloon is 72
meters high.
(
Hint
: remember 345 right triangles.)
1.
rate = 50 meters/min
2.
rate = 48 meters/min
correct
3.
rate = 49 meters/min
4.
rate = 51 meters/min
5.
rate = 52 meters/min
Explanation:
Let
h
be the height of the balloon
t
seconds
after it is released and
s
the distance of the
balloon from the observer as shown in the
figure
54
balloon
h
s
observer
Then by Pythagoras’ theorem,
h
2
+ 54
2
=
s
2
.
Differentiating this equation implicitly with
respect to
t
we see that
2
h
dh
dt
= 2
s
ds
dt
,
i.e.
,
ds
dt
=
h
s
dh
dt
.
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 Fall '08
 schultz
 Derivative, Differential Calculus, Limit of a function, Hamrick

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