MIXTURE PROBLEM 2:  A tank holds 100 gallons of liquid.  The tank is half full with a salt water solution that contains 0.1 lb of salt per gallon.  Pure water is added to the container at the rate of 2 gallons per minute and at the same time one gallon of salt water per minute is removed from the tank.  Assume that the concentration of salt in the water in the tank remains uniform throughout.  When the tank becomes full it begins to overflow and at that time a total of 2 gallons per minute of salt water will be leaving the tank.  Construct a piecewise continuous function S(t) that gives the amount of salt in the tank as a function of time t where t = 0 represents the time when the 2 gallons per minute of pure water began being added to the tank and the 1 gallon per minute of "well-stirred, uniform" mixture began exiting the tank.  The picture at the right shows the amount of salt in the tank as a function of time.

I guess we could say that this is a thermometer that reacts pretty slowly to an increase in the temperature around it.

Two minutes later the milk diluted hot chocolate will have a temperature of 160^oF if it is in a room whose temperature is 80^oF.  Use this fact to calculate k.  (a)  If the milk was added to the hot chocolate at time t=0 minutes, how long will it take for the temperature of the milk diluted hot chocolate to reach 120^oF?  (b)  Suppose instead that the milk is not added to the hot chocolate until the originally 200^oF hot chocolate has been in the 80^oF room for 5 minutes.  In this case how long will it take (total) for the hot chocolate (ultimately diluted by milk) to reach a temperature of 120^oF?  (c)  Using the 80^oF room and the 50^oF milk, how could the hot chocolate be cooled to 120^oF in the least amount of time and what would this time be? 

Red gives the temperature curve when the milk is added immediately.

Blue gives the temperature curve when the milk is not added until five minutes after the hot chocolate was poured into the cup in the 80^oF room.

Green gives the temperature curve when the milk is added when the temperature of the hot chocolate reaches 127.778^oF.

The graph above is a zoom in on the graph at the left around the points where the three temperature curves are intersecting the horizontal line along which the temperature equals 120^oF.