Separable Differential Equations

Question

A 55 "-gallon" tank is filled with brine (water nearly saturated with salt; used as a preservative) holding 12 ⁢ lb of salt in solution. A salt solution containing 0.4lb of salt per gallon is added to the tank at the rate of 1 gal/min. The contents of the tank are continuously and thoroughly mixed and drained out at the same rate. What is the amount of salt in the tank after an hour? Round any intermediate calculations, if needed, to no less than six decimal places, and round your final answer to three decimal places.

Answer 1

The equation for the accumulation of mass of salt in the tank is :

dm/dt = .4 lb/gal * 1 gal/min - (m lb/ 55 gal) * (1gal/min)

Separating is one way to solve this differential equation.

dm/(.4 - m/55) = dt You can integrate using definite integral or indefinite and solve for C using m(0) = 12 lb

-55ln((.4 - m/55)/(.4 - 12/55)) = t (note t needs to be in minutes

Algebra ensues: m = -55((.4 - 12/55)e^-60/55 -.4)

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