A \(500\)-liter tank has \(100\)-liters of pure water when a process begins that pumps in saltwater with \(30\) grams per liter of salt at \(4\) liters per minute, while the liquid in the tank is kept thoroughly mixed and allowed to drain at \(2\) liters per minute. The process automatically stops when the tank reaches capacity.

(a)
Let’s have \(y(t)\) be the grams of salt in the tank, at time \(t\geq 0\) minutes into the process. Let’s derive an IVP that \(y(t)\) must satisfy.
(b)
Let’s solve this IVP.
(c)
Let’s determine the amount of salt in the tank when the process stops, and the concentration in grams per liter.

A financial planner is modeling a client’s portfolio of financial assets. They assume that the long-run rate of return will always be at least equivalent to a constant annualized rate of return of \(r\) %, that the initial principle is \(A_0\), and that there will be a fixed annual withdrawal rate of \(N+\alpha t\) where \(N,\alpha \) are fixed positive constants.

(a)
Let \(y(t)\) be the model’s worst case value for the portfolio after \(t\geq 0\) years. What IVP must \(y(t)\) satisfy?
(b)
Let’s determine the general solution.
(c)
What must \(A_0\) satisfy to guarantee an exponential growth term in \(y\)?