(b) A 10 kL tank of water provides water for a small village. Water is drunk from the tank at a rate of 1000 L per day, and the water is replenished at the same rate. Unfortunately, the filter/purifier for the intake water is malfunctioning. Conse- quentially the water that is replenishing the tank is contaminated; it contains 0.00125 mg/L of heavy metals. Suppose that when the contamination was discovered the tank was 3/4 full and contained a, mg of heavy metals. (i) Find a formula, A(t), for the amount of heavy metals in the tank at time t. Hint: make sure your units are consistent for all quantities. We suggest mg/kL If the concentration of heavy metals in the water is below 0.001 mg/L then the water is still safe to drink. You may assume that the heavy metals in the tank are well mixed (and thus evenly distributed) in the tank. The maintenance company has been notified, and have indicated that the they'll be on premises to service the filter/purifier in 7 days (and please make sure some- body is on premises during that entire day). However, we know from experience that they can be anywhere up to 3 days late. (ii) If the initial amount of heavy metals in the tank was 2 mg, will the concen- tration reach dangerous levels before the company services the tank? (Yes, no, or maybe?). Justify your answer. (iii) What is the largest initial amount of heavy metals that will definitely not reach dangerous levels before the company services the tank?

3. A large tank with the capacity of 500 gallons is fully filled with pure water. Brine containing 2 pounds of salt per gallons is pumped into the tank at a rate of 5 gal/min. Find the amount A(t) pounds of salt in the tank at timer minutes, if the well-mixed solution is pumped out (a) with the same rate as the entering one; and (b) with the rate of 10 gal/min. In case (a), how much salt is in the tank after a long time? In case (b), what is the concentration of the solution in the tank by the time it is empty? (c) Let everything be the same except that, initially, the tank is filled with 250 gallons of pure water, while the well-stirred solution is pumped out with the flow rate of 2.5 gal/min. Find (t). What is the concentration of the solution in the tank by the time it is fully filled? Hint: (a) The volume of the solution in the tank at time (b) The volume of the solution in the tank at time minutes is V(t)= 500 gallons minutes is V(t)=500-5r gallons (c) The volume of the solution…

5. Suppose that sodium pentobarbital is used to anesthetize a dog. The dog is anesthetized when its bloodstream contains at least 45 milligrams (mg) of sodium pentobarbitol per kilogram of the dog's body weight. Suppose also that sodium pentobarbitol is eliminated from the dog's bloodstream at a rate which is proportional to the amount of sodium pentobarbitol in the bloodstream and that after 5 hours half of the initially prescribed sodium pentobarbitol is elliminated from the bloodstream. Set up and solve an initial value differential equation. What single dose should be administered in order to anesthetize a 50-kg dog for 1 hr?