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1. Homework Statement
The idea of the problem is to find out how long it would take to flush the Great Lakes of pollution. They're set up as a series of five tanks and you are given inflow rates of clean water, inflow rates from the other tanks, and outflow rates. See attached figure.
2. Homework Equations
dA/dt=rateinrate out where A is the amount of pollution at time t
3. The Attempt at a Solution
first I wrote equations for each lake. rather than using A as my variable, I used the first letter of each lake (with n for Ontario) to stand for the amount of pollution in the given lake at time t. this gives...
ds/dt=15s/2900
dm/dt= 38m/1180
dh/dt=15s/2900 + 38m/1180  68h/850
de/dt=68h/850  85e/116
dn/dt=85e/116  99n/393
rearranging and pulling out the differential operator leads to the following system
(D + 38/1180)[m]=0
(D  68/850)[h] + 38m/1180  15s/2900=0
(D + 15/2900)=0
(D + 85/116)[e] 68h/850=0
(D + 99/393)[n]  85e/116=0
and there you have it. A system of five equations with five variables. It seems like it should be fairly straightforward to solve from here, but I can't figure out what to do next. I'm stuck!!
The idea of the problem is to find out how long it would take to flush the Great Lakes of pollution. They're set up as a series of five tanks and you are given inflow rates of clean water, inflow rates from the other tanks, and outflow rates. See attached figure.
2. Homework Equations
dA/dt=rateinrate out where A is the amount of pollution at time t
3. The Attempt at a Solution
first I wrote equations for each lake. rather than using A as my variable, I used the first letter of each lake (with n for Ontario) to stand for the amount of pollution in the given lake at time t. this gives...
ds/dt=15s/2900
dm/dt= 38m/1180
dh/dt=15s/2900 + 38m/1180  68h/850
de/dt=68h/850  85e/116
dn/dt=85e/116  99n/393
rearranging and pulling out the differential operator leads to the following system
(D + 38/1180)[m]=0
(D  68/850)[h] + 38m/1180  15s/2900=0
(D + 15/2900)
(D + 85/116)[e] 68h/850=0
(D + 99/393)[n]  85e/116=0
and there you have it. A system of five equations with five variables. It seems like it should be fairly straightforward to solve from here, but I can't figure out what to do next. I'm stuck!!
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