Question: differential equations mixing problems for all problems you must include...
Differential Equations: Mixing Problems
(FOR ALL PROBLEMS, YOU MUST INCLUDE ANY IMPORTANT SUPPORTING IDEAS/ WORK THAT SUPPORT YOUR CONCLUSIONS. IF YOU USE A THEOREM OR FORMULA, PLEASE INDICATE THAT YOU'RE DOING SO) - (PLEASE, WRITE NEAT)
1. A tank initially contains 100 liters of pure water. Saline solution containing 9 grams of salt per liter is mixed into the tank at a constant rate of 3 liters per minute. The tank is perfectly mixed, and the resulting mixure flows out of the tank at the constant rate of 4 liters per minute.
(a) When is the tank empty?
(b) Find the amount of salt in the tank after t minutes.
(c) At what time is the amount of salt in the tank maximized?
2. Lake Bulldog, a popluar swimming lake located somewhere in northern Minnesota, holds 4, 000, 000 liters of water, and has one creek flowing in, at a constant rate of 20 liters per minute, and one creek flowing out, also at a constant rate of 20 liters per minute. A factory was recently built upstream from Lake Bulldog, which dumps highly phosphoric water into the inflowing creek. Since this began, the inflowing creek has seen a phospho- rous concentration of .1 milligrams per liter (mg/L), a dramatic jump in concetration. Lake Bulldog initally has a concentration of .01 mg/L of phosphrous (a normal concen- tration for an unpolluted lake). If .02 mg/L is the threshold where the lake no longer is safe for swimming, for how many days (after the factory began operation) can swimmers enjoy the lake before it becomes too toxic?