Mathematics: A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well-mixed solution is pumped out at the same rate. Find the number A(t) of pounds of salt in the tank at time t.

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A large tank is filled to capacity with 500 gallons

tylorroundy

20.01.2022

There's no salt in the tank at the start, so A(0) = 0.

Salt flows in at a rate of

(2 lb/gal) * (5 gal/min) = 10 lb/min

and flows out at a rate of

(A(t)/500 lb/gal) * (5 gal/min) = A(t)/100 lb/min

Then the net flow rate is given by

A'(t) = 10 - A(t)/100

in gal/min.

Solve the ODE:

$A'(t)+\dfrac{A(t)}{100}=10$

$e^{t/100}A'(t)+e^{t/100}\dfrac{A(t)}{100}=10e^{t/100}$

$\left(e^{t/100}A(t)\right)'=10e^{t/100}$

$e^{t/100}A(t)=1000e^{t/100}+C$

$A(t)=1000+Ce^{-t/100}$

With A(0) = 0, the above gives us 0 = 1000 + C, so that C = -1000, and the particular solution to this IVP is

$A(t)=1000(1-e^{-t/100})$

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