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Unformatted text preview: 1 Solutions 27 and simplifying gives
. The initial condition
implies
so
for
. At
we get
lbs salt. Once the tank is full, the inﬂow
and outﬂow rates will be equal and the brine in the tank will (in the
limit as
) stabilize to the concentration of the incoming brine, i.e.,
lb/gal. Since the tank holds 100 gal, the total amount present will
approach
lbs. Thus
.
31. input rate: input rate
output rate: output rate Let
denote the amount of pollutant at time
. Since
input rate output rate it follows that
is a solution of the initial
value problem Rewriting this equation in standard form gives the diﬀerential equation
. The coeﬃcient function is
and the integrating
factor is
. Thus
. Integrating and simplifying
gives
where is the constant of integration. The initial
condition
implies
so
(a)
(b) When the river is cleaned up at
we assume the input concentration is
. The amount of pollutant is therefore given by
This will reduce by
when
. We solve the
equation
for and get
. Similarly, the pollutant
will reduce by
when
.
(c) Letting
Lake Erie:
Lake Ontario: and be given as stated for each lake gives:
years,
years, years.
years 32. Let
and
denote the amount of salt in Tank 1 and Tank 2,
respectively, at time .
L
g
g
input rate for Tank 1: input rate
.
min
L
min
L
g
g
output rate for Tank 1: output rate
.
min
L
m
The initial value problem for Tank 1 is thus: Simplifying and putting this equation in standard form gives
. The integrating factor is
. Thus . ...
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 Fall '08
 BELL,D

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