Two tanks are connected as in Figure 1.6. Tank 1 initially contains 20 pounds of salt dissolved in 100 gallons of brine. Tank 2 initially contains 150 gallons of brine in which 90 pounds of salt are dissolved. At time zero, a brine solution containing 1/2 pound of salt per gallon is added to tank I at the rate of 5 gallons per minute. Tank 1 has an output that discharges brine into tank 2 at the rate of 5 gallons per minute, and tank 2 also has an output of 5 gallons per minute. Deter- mine the amount of salt in each tank at any time. Also, determine when the concentration of salt in tank 2 is a minimum and how much salt is in the tank at that time. Hint: Solve for the amount of salt in tank 1 at time t and use this solution to help determine the amount in tank 2. S pal min: 12 Ih gal S gil tain Tauk

Consider three connected lakes, A, B, and C, where lake B has a volume of 2650 km3, and water flows from lake A into lake B, and from lake B into lake C, both inflows and outflows at the same rate of 525 km3/year. Suppose that at time t=0 (years), the pollutant concentration of lake B, caused by past industrial pollution that has now been ordered to cease, is five times that of lake A. If the outflow from lake B to lake C is perfectly mixed lake water, how long will it take to reduce the pollution concentration in lake B to twice that of lake A? Choose all correct answers. B Approximately 6.3 years. Approximately 4.4 years. Approximately 5 years. Approximately 8.3 years. Approximately 7 years.

Part 1 Homing pigeons avoid flying over water. Suppose a homing pigeon is released on an island at point​ C, which is 7 mi directly out in the water from a point B on shore. Point B is 16 mi downshore from the​ pigeon's home loft at point A. Assume that a pigeon flying over water uses energy at a rate 1.23 times the rate over land. Toward what point S downshore from A should the pigeon fly in order to minimize the total energy required to get to the home loft at​ A?

1. Consider two interconnected tanks as shown below. Tank 1 initially contains 600 gal of water with 3 lb of salt, and Tank 2 initially contains 400 gal of water with 1 lb of salt. Water containing 0.4 lb/gal of salt flows into Tank 1 at a rate of 3 gal/min. The well stirred mixture in Tank 1 flows into Tank 2 at a rate of 4 gal/min. The well stirred mixture in Tank 2 leaves the tank at a rate of 6 gal/min and additionally, flows back into Tank 1 at a rate of 2 gal/min. Let Q₁(t) and Q₂(t) denote the amounts of salt (in lb) in Tank 1 and Tank 2 at time t in minutes. Let V₁ (t) and V₂(t) denote the volumes of solutions (in gal) in Tank 1 and Tank 2 at time t in minutes. Set up the initial value problem satisfied by Q₁(t) and Q₂(t) describing the mixing process. (Show your work to get a full credit) Tank 1 72 Tank 2 T4