MATH 33B Lecture 7: Math 33B Dis 3_ Mixing Problems, Exact ODEs (01/24/17)

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

Suppose that a tank initially contains 25 gallons of water with a concentration of 2 grams per gallon of salt. To gradually flush the salt out of the tank, you turn on a pipe which pours pure water into the tank at a rate of 5 gallons per hour. Suppose that a drain at the bottom of the tank opens 4. simultaneously, at which the well-stirred tank water leaves at a rate of 3 gallons per hour (a) 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 Ax (b) By solving your DE and using the initial condition to find the undetermined constant, compute an exact formula for X(t). Use your formula to determine the number of days it will take for the amount of salt in the tank to fall under 1 gram. (Note that t is in units of hours in your equations.) You can take for granted that the tank is large enough to hold the increased volume of water

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 fows othughapipe at a rate of 5 gallons per hour. Supposha the tank can hold (a) It should be clear that the volume of water in the tank is increasing as time passes. At what time 100 gallons of water. 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 t 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 S t