|Feb12-07, 03:09 AM||#1|
differential equations - mixing problem (more complicated)
A 500 gallon tank originally contains 100 gallons of fresh water. Beginning at time t=0, water containing 50 percent pollutants flows into the tank at the rate of 2 gal/min and the well stirred solution leaves at the rate of 1 gal/min. Find the concentration of pollutants in the tank at the moment it overflows.
The answer is 48%
my attempt, something is wrong as i did not get that answer or anywhere near it.
rate in: 2 x 0.50
rate out: 1 x s(t)/(100+t)
s'(t) = 1- s(t)/(100+t)
s'(t) + s(t)/(100+t) = 1
a(t) = 1/(100+t) , b(t) = 1
using u(t) = exp(integ(a(t)dt)):
u(t) = exp(integ(a(t)dt))
u(t) = exp (ln (100+t))
u(t) = 100+t
using d/dt u(t)s(t) = u(t)b(t):
d/dt [(100+t)s(t)]= (100+t)(1)
(100+t)s(t) = 100t + 1/2t^2 + C
s(t) = [100t + 1/2t^2 +C ]/(100+t)
sub s(0) = 0 into s(t) [is this even correct?]
0 = [0+0+c]/[100+0]
s(t) = [100t + 1/2t^2]/(100+t)
c(t) = s(t)/100
c(t) = [100t + 1/2t^2]/100(100+t)
then i found when the tank overflow:
500 = 100+t
then found c(t)
c(400) = [100(400)+1/2(400)^2]/100(100+400)
c(400) = 2.4
which is wrong.
can anyone help me?
|Feb12-07, 06:44 AM||#2|
Okay, good so far.
|Feb12-07, 06:48 PM||#3|
thanks very much
that was a stupid mistake on my part!
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