3. Consider a semi-permeable membrane separating two chambers, each of which contains a liter of salt-water. Let A(t) and B(t) denote the amount salt (in grams) after t minutes. The rate

at which salt moves from Chamber A to Chamber B (in grams per minute) is proportional to the difference in concentration; in other words, it can be written as k(A(t)- B(t)) for some constant k. (a) Write formulas expressing A'(t) and B' (t) in terms of A(t), B(t) and k. (b) Should the constant k be positive or negative, in order for this situation to make physical sense? (e) Suppose that A(0) = 5 grams and B(0) = 7 grams. What can we say about A(t) + B(t)? Why? (d) Setting k = 1 and using the initial values from part (e), find explicit formulas for A(t) and B(t). (Hint: Use your previous answer to forget about B(t) and write everything in terms of A(t).) (e) What happens to A(t) and B(t) after a very long time? Explain how you could have predicted this without explicitly solving the differential equation.