How to solve for the arc CFD? (Picture)
1 answer:
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Until the concerns I raised in the comments are resolved, you can still set up the differential equation that gives the amount of salt within the tank over time. Call it 
.
Then the ODE representing the change in the amount of salt over time is
and this with the initial condition
You have
![\dfrac{\mathrm d}{\mathrm dt}\left[e^{t/250}A(t)\right]=\dfrac25e^{t/250}(1+\cos t)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dt%7D%5Cleft%5Be%5E%7Bt%2F250%7DA%28t%29%5Cright%5D%3D%5Cdfrac25e%5E%7Bt%2F250%7D%281%2B%5Ccos%20t%29)
Integrating both sides gives
Since, you get
so the amount of salt at any given time in the tank is
The tank will never overflow, since the same amount of solution flows into the tank as it does out of the tank, so with the given conditions it's not possible to answer the question.
However, you can make some observations about end behavior. As
, the exponential term vanishes and the amount of salt in the tank will oscillate between a maximum of about 100.4 lbs and a minimum of 99.6 lbs.
Then the ODE representing the change in the amount of salt over time is
and this with the initial condition
You have
Integrating both sides gives
Since
so the amount of salt at any given time in the tank is
The tank will never overflow, since the same amount of solution flows into the tank as it does out of the tank, so with the given conditions it's not possible to answer the question.
However, you can make some observations about end behavior. As