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We have 40 grams of salt that has been dissolved in a tank containing 300 liters of water. Brine containing 2g of salt per liter is being pumped into the tank at a rate of 2 liters per minute and then the mixed solution is pumped out at the same rate. Find the number of grams of salt A(t) after 30 minutes. The Mathematics behind the Model: Mixing problems like this are gov- erned by the concept that the rate of change of the amount of salt is equal to the rate of salt entering minus the rate of salt leaving. Let A(t) be the amount of salt at time t and write dA dt = (rate entering) (rate leaving) := Rin - Rout - 1. (5 points) Recall that the rate entering is (concentration in) (fluid rate in) and write the expression for Rin for this problem. Be sure to include units to check that you have the correct quality for this rate. 2. (5 points) Recall that the rate leaving is (concentration leaving)-(fluid rate leaving) and write the expression for Rout for this problem. Be sure to include units to check that you have the correct quality for this rate. 3. (5 points) Write the differential equation that models this problem. 4. (5 points) What is the dependent variable? 5. (5 points) What is the initial condition? 6. (5 points) Write the (linear) differential equation in standard form. 7. (5 points) Solve the differential equation, interpret the initial condition, and give the function A(t) that gives the number of grams of salt at anytime. 8. (5 points) Find, to the nearest gram, the number of grams of salt A(t) after 30 minutes. We have 40 grams of salt that has been dissolved in a tank containing 300 liters of water. Brine containing 2g of salt per liter is being pumped into the tank at a rate of 2 liters per minute and then the mixed solution is pumped out at the same rate. Find the number of grams of salt A(t) after 30 minutes. The Mathematics behind the Model: Mixing problems like this are gov- erned by the concept that the rate of change of the amount of salt is equal to the rate of salt entering minus the rate of salt leaving. Let A(t) be the amount of salt at time t and write dA dt = (rate entering) (rate leaving) := Rin - Rout - 1. (5 points) Recall that the rate entering is (concentration in) (fluid rate in) and write the expression for Rin for this problem. Be sure to include units to check that you have the correct quality for this rate. 2. (5 points) Recall that the rate leaving is (concentration leaving)-(fluid rate leaving) and write the expression for Rout for this problem. Be sure to include units to check that you have the correct quality for this rate. 3. (5 points) Write the differential equation that models this problem. 4. (5 points) What is the dependent variable? 5. (5 points) What is the initial condition? 6. (5 points) Write the (linear) differential equation in standard form. 7. (5 points) Solve the differential equation, interpret the initial condition, and give the function A(t) that gives the number of grams of salt at anytime. 8. (5 points) Find, to the nearest gram, the number of grams of salt A(t) after 30 minutes.
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