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.

Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

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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21.
16. A solution is made that contains 9 pounds of salt per every 10 gallons and flows into a half tank of water that holds 2021 gallons at time 3D0. Then the solution enters the tank from one side at a rate of 12 gallons per minute and exits the tank from the other side at a rate of 11 gallons per minute. a. How much salt is in the tank when full?
2. A small lake of volume 10 cubic meters has accidentally been polluted by 10 000 kilograms of a highly toxic substance. A river flows into and out of the lake at a rate of 20000 cubic meters per hour. Assuming that the entering river contains fresh water and that the toxic substance is fully mixed throughout the lake at all times, find the number of hours for the mass of the pollutant to decrease to an acceptable 100 kilograms.
Number 20-
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
Brine containing 2lbs of salt per gallon runs into a tank at 2 gal/min, brine from this tank runs into a second tank at 3 gal/min and brine runs out of the second tank at 3 gal/min. Initially, the first tank conditions 10 gal of brine with 30 lb of salt in solution and the second tank 10 gal of fresh water. Assuming uniform consideration in each tank, find the quantity of salt in the second tank at the end of 5min.
3. A tank contains 400 liters of brine holding 300 kg of salt in solution. Water containing 125 g of salt per liter flows into the tank at the rate of 12 liters per minute, and the mixture kept uniform by stirring, flows out at the same rate. Find the among of salt at the end of 90 minutes. Hint: = Pinin dt - Tout s+ (Fin-Tout))
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
a tank in which 1 gram of chlorine is initially present in 80.38 m3 of a solution of water and chlorine. A chlorine solution concentrated at 1.49 g/m3 flows into the tank at a rate of 3 m3 /min, while the uniformly mixed solution exits the tank at 7 m3 /min. What will be the amount of chlorine present in the tank after 1 minute?

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