Atank contains 100 gal of fresh water. a solution containing 2 lb/gal of soluble lawn fertilizer runs into the tank at the rate of 1 gal/min, and the mixture is pumped out of the tank at the rate of find the maximum amount of fertilizer in the tank and the time required to reach the maximum.

Let be the amount of fertilizer in the tank at time . Then




This ODE is separable, and you can write



Integrating both sides gives








At , there is no fertilizer in the tank, so . This means



so that the amount of fertilizer in the tank is given by



This function is constantly increasing, so it never achieves its maximum, but there is a strict upper bound as . You have lbs of fertilizer in the tank, which makes sense. Given enough time, the tank will be completely saturated with the fertilizer solution, so 2 lb/gal*100 gal = 200 lbs.

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