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MAP 2302
Fall2011
SOLUTIONS:
Quiz 3
1. A swimming pool whose volume is 10,000 gal contains water that is
0.01% chlorine. Starting at
t
= 0 city water containing 0.001% chlorine is
pumped into the pool at a rate of 5 gal/min. The pool water ﬂows out at the
same rate. What is the percentage of chlorine in the pool after 1 h? When
will the pool water be 0.002% chlorine?
SOLUTION. Let
x
(
t
) be the amount of chlorine in the pool at time
t
.
Then the concentration of chlorine is
c
(
t
) =
x
(
t
)10

4
.
The input rate is 5
×
0
.
001
100
= 5
×
10

5
.
The output rate is 5
×
c
(
t
) = 5
×
10

4
x
(
t
).
The initial condition is
x
(0) =
c
(0)10
4
=
0
.
01
100
10
4
= 1.
The problem is to ﬁnd
c
(60) in percents and to ﬁnd
t
such that
x
(
t
)
10
4
=
0
.
002
/
100. Remember, 1 h = 60 minutes.
The initial value problem is
dx
dt
= 5
×
10

5

5
×
10

4
x x
(0) = 1
.
The equation is separable. It can be rewritten as
dx
x

10

1
=

5
×
10

4
dt.
The integration of both sides gives us ln
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 Spring '08
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