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
80 bacteria after 1 hour, how many will there be after 100 minutes?
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 Fall '08
 CHESTKOFSKY
 Math, Calculus, Derivative, 1 Hour, Logarithm, 20 km

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