An 80 gallon tank initially contains 40 gallons of water wit

An 80 gallon tank initially contains 40 gallons of water with 10 lbs of salt dissolved in it. At time t = 0; brine containing 2 lbs of salt per gallon begins to be pumped into the tank at a rate of 4 gallons per minute. Also at t = 0; a drain is opened at the bottom of the tank, that lets out 2 gallons of well-mixed brine per minute.

(a) Find an explicit expression for the amount of salt in the tank after t minutes, and the amount of salt in the tank at the moment it is completely full.

(b) Now, suppose that at the moment the tank is completely full, the incoming supply of brine is shut off so that no further brine enters the tank, and pure water is pumped in instead, at a rate of 1 gallon per minute. The drain remains open as before, letting out 2 gallons per minute. Find an expression for the amount of salt in the tank at any time after the incoming supply is switched to pure water, and find the amount of salt in the tank one hour after the pumping operation initially began.

Solution

(a)

let\'s the function be

s(t) = kgs of salt in the tank at time t minutes.

and we know that

s\'(t) = (rate of salt going in) ? (rate of salt going out).

First, the