Single-compartment model with changing volume. Solve (a), (b), and (c).
In all of the following problems, be careful to use a letter other than C for your undetermined constants, as C is reserved for the concentration C(t) in this context Suppose that a tank initially contains 50 gallons of pure water. Water containing a concentration of 3 grams per gallon of salt begins to enter the tank at a rate of 6 gallons per hour, while the well-stirred tank water flows out through a pipe at a rate of 5 gallons per hour. Suppose that the tank can hold 3. 100 gallons of water (a) It should be clear that the volume of water in the tank is increasing as time passes. At what time will the tank fill up? (b) Write down a differential equation which describes the amount X(t) of salt in the tank at any time t 2 0, together with an initial condition satisfied by X. (Note that your equation will only model the situation at hand up until the t-value you found in (a).) (c) Let tf denote the time at which the tank becomes full (which you computed in (a)). By solving the DE you wrote down in (b) and using the initial condition to find the undetermined constant, compute an exact formula for X(t) which is valid for 0 St
