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Fall 2009 Math 113 Exam #2 Solutions
1. An asteroid hits the Atlantic Ocean and creates an expanding circular wave. If the area enclosed by
this wave increases at the rate of 200 km
2
/min, how fast is the
diameter
of the wave expanding when
its
radius
is 20 km?
Answer:
We know that the area enclosed by the wave is given by
A
(
t
) =
πr
(
t
)
2
and that the diameter
is 2
r
(
t
). From this latter, the rate of change of the diameter (which is what we’re trying to determine)
is 2
r
0
(
t
).
Diﬀerentiating the expression for
A
,
A
0
(
t
) = 2
πr
(
t
)
r
0
(
t
)
.
Therefore,
r
0
(
t
) =
A
0
(
t
)
2
πr
(
t
)
.
At the time
t
0
that we’re interested in,
A
0
(
t
0
) = 200 and
r
(
t
0
) = 20. Plugging these values into the
above equation, we see that
r
0
(
t
0
) =
200
2
π
(20)
=
200
40
π
=
5
π
.
Therefore, at this instant the diameter is increasing at a rate of
2
r
0
(
t
0
) = 2
5
π
=
10
π
km/min
.
2. Suppose 10 bacteria are left to grow in a petri dish. At least for the ﬁrst few hours, the rate of
population growth of this bacterial culture will be proportional to the size of the culture. If there are
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This note was uploaded on 01/18/2012 for the course MATH 2250 taught by Professor Chestkofsky during the Fall '08 term at University of Georgia Athens.
 Fall '08
 CHESTKOFSKY
 Math, Calculus

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