Question

2. (10 points) An aquarium has a 1000 L tank containing 400 L of salt water with a concentra- tion of 210 grams per liter. A salt water solution with a concentration of 470 grams per liter is pumped into the tank at a rate of 5 liters per minute. The well-mixed solution is drained from the tank at a rate of 3 liters per minute. Additionally, the tank is uncovered, so fresh water evaporates from the tank at a rate of 0.5 liters per minute. Let y(t) be the number of grams of salt in the tank t minutes after the pumping begins. (a) Find the domain of y(t). Hint: Think about the volume of the saltwater solution in the tank. We do not want the tank to overflow. (b) Set up, but DO NOT SOLVE, an initial value problem which describes the dynamics of this system. That is, find a formula fordu in terms of y and/or t, AND specify a known point on the solution curve. Be sure you clearly explain where your differential equation comes from. Hints: 1. Think about the rate that salt is entering the tank and the rate that salt is leaving the tank. 2. The units of are grams of salt per minute. You should make sure each term in your differential equation has the same units. 3. When water evaporates, it does not take any salt with it. This may seem quite hard the first time you do one of these, and this one is on the hard side, but we will practice more of them later in Chapter 10. This particular problem is from one of my old 16B exams. The ODE can be solved using the method of integrating factors from 10.2, but this question does not ask you to solve it.

Answer 1

tc dt cen hn h Lip