Suppose two interconnected tanks partially filled with brine. Pure water flows into tank 1 at arate of 2 liters per minute. Pure water flows into tank 2 at a rate of 3 liters per minute. Brineflows out of tank 2 and into tank 1 at a rate of A liters per minute. Brine flows out of tank 1and into a pipe at a rate of B liters per minute. Then C liters of the brine in this pipe exitvia a drain in this pipe every minute. The remaining B – C liters of brine enter tank 2 everyminute. Assume uniform mixing at all times t.(a) Let A1 (t) be the amount of salt in tank 1 at timet > 0. Let A2(t) be the amount of saltin tank 2 at time t > 0. Construct a system of differential equations to model the rate ofchange of salt in both tanks at time t > 0.(b) Set up a matrix equation to solve this linear system of ODE’s. What must be true aboutA, B, and C so that the entries of the matrix are constant?(c) Choose values of A, B, and C so that the entries of the matrix in part (b) are constant.Then solve the IVP.(d) Is it possible to model the relationship between A1 and A2 using a constant-coefficient,second order ODE? If so, write down this ODE. If not, explain why and give an exampleof a constant-coefficient, second order ODE and show how you would transform it into amatrix equation.

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Answered: Suppose two interconnected tanks…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider three connected lakes, A, B, and C, where lake B has a volume of 2650 km, 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 km2/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) OA More than five years. OB Less than eight years. OC Less than six years. D D. More than 11 years.
Three tanks connected by pipes are shown on a figure below. Originally they contain 10, 20, and 40 gallons of pure water respectively. A brine with salt concentartion of 11b/gal flows into the first tank with the rate 5 gal/min and a well-mixed solution flows from the first to the second tank with the rate 8 gal/min. From the second tank the well-mixed solution flows into the third tank with the rate 8 gal/min and part of a well-mixed solution flows from the third to the first tank with the rate 3 gal/min while another part flows outside the system with the rate 5 gal/min. Write down the sys- tem of differential equations describing the amount of salt in each tank and put it into the matrix form. mixture 11b/gal flow 5 gal/min 10 gal 8 gal/min J 20 gal 3 gal/min 8 gal/min 40 gal 5 gal/min
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1. Mr. Pein has an aquarium that has a volume of 2000 mL and contains freshwater fish. One day, Mr. Pein's son accidentally spilled 1 kilogram of salt into the aquarium. The fish will die if they are in water with a salt concentration of more than 10-3 grams/mL for more than 30 minutes. So that the fish in the aquarium do not die, Mr. Pein must immediately reduce the salt concentration in the aquarium to 10-3 grams/mL by flowing fresh water at a constant rate through the aquarium, and removing the water from the aquarium at the same rate. What is the minimum water speed so that the fish don't die? In 500 А. 60000 mL/minute 2000 В. 30 In 500 mL/min 1996 С. mL/minute 30 In 500 D. 6000 mL/minute 200 Е. 30 In 500 mL/min
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.
What geometric transformation of 3D space happens in this matrix-vector multiplication? 010 -1 0 0 0 01 9).( (3) Please note: All rotation directions to be understood as when looking along the rotation axis towards the origin. O Clockwise rotation by 90 degrees around the x3 axis Counter-clockwise rotation by 90 degrees around the x3 axis Clockwise rotation by 90 degrees around the x₂ axis Counter-clockwise rotation by 90 degrees around the x2 axis O Clockwise rotation by 90 degrees around the x₁ axis Counter-clockwise rotation by 90 degrees around the x₁ axis O Mirroring across the x₁-x2 plane Mirroring across the x₁-x3 plane Mirroring across the X₂-X3 plane
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…
9. A glass container can hold 35 liters of water. It currently has 10 liters of water with 15 grams of Gatorade power initially dissolved in the container. A solution is poured into the container at 3 liters per minute - the solution being poured in has 0.5 grams per liter of Gatorade powder. Assume the solution in the container is well mixed. There is an outflow at the bottom of the container which has liquid leaving at 1 liter per minute. Let G(t) denote the amount of Gatorade powder in the tank at time t. a. Setup the differential equation for G'(x). b. Solve for the general solutiont) c. Use initial condition to find the specific solution. (Write out the entire solution, with the constant(s) plugged in.) ) d. When will the container overflow?
2. A large tank initially contains 100 gallons of brine in which 10 lb of salt is dissolved. Starting at t = 0, pure water flows into the tank at the rate of 5 gal/min. The mixture is kept uniform by stirring, and the well-stirred mixture simultaneously flows out at the slower rate of 2 gal/min. a) How much salt is in that tank at the end of 15 min and what is the concentration at that time? b) If the capacity of the tank is 250 gallons, what is the concentration at the instant the tank overflows?
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?
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

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