Two tanks are interconnected as illustrated below:Tank 12NgalmingalminTank 2Suppose that Tank 1 contains 1 gallon of brine in which 4 pounds of salt are initially dissolved,and Tank 2 initially contains 1 gallon of pure water. At time t 0, the mixtures are pumpedbetween the two tanks, each at a rate of 2 gal/min. Assume the mixtures are well stirred. Letx₁ (t) and x₂ (t) be the amount of salt (in pounds) in Tank 1 and Tank 2, respectively, at timet.a. Formulate the system of linear differential equations that describes the amount of salt inthe tanks.b. Obtain the initial-value problem above in the matrix form X'c. Examine the solution of the system in part (b) for x₁ and x₂.d. Compute the amount of salt in each tank after 45 seconds.=AX.

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Answered: Two tanks are interconnected as…

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 the cascade of two tanks of brine shown in the figure below, with V1=99 gallons and V2=204 gallons being the volumes of brine in the two tanks. Each tank also initially contains 51 lb of dissolved salt. The three flow rates indicated in the figure are each 4.6 gal/min, with pure water flowing into tank 1. (a) Find the amount x(t) of salt in tank 1 at time t. x(t)=? (b) Suppose that y(t), the amount of salt in tank 2 at time t, is given by the differential equation dy/dt=(4.6x/99)−(4.6y/204). Solve for y(t) using the function x(t) found in part (a) y(t)=?
Set up an I.V.P. and solve using the following: Suppose a large mixing tank initially holds 500 liters of pure water. A brine solution is pumped into the tank at a rate of 3 L/min with a concentration of 0.2 kilograms of salt per liter. If the well-stirred solution is pumped out at the same rate, find the number of kilograms of salt in the tank at time t, A(t).
When a vertical beam of light passes through a transparent medium, the rate at which its intensity I decreases is proportional to I(t), where t represents the thickness of the medium (in feet). In clear seawater, the intensity 3 feet below the surface is 25% of the initial intensity I of the incident beam. Find the constant of proportionality k, where dI dt k= In(4) 3 = KI. What is the intensity of the beam 17 feet below the surface? (Give your answer in terms of Io. Round any constants or coefficients to five decimal places.) 0.000761 X
A pasta company with a blue box produces 1,400 tons of semolina pasta per day. The pasta is fed into a machine as a large, loose dough and then squeezed through a bronze die to form a desired shape. As the pasta is squeezed through the die, the volume of space it takes up decreases while the mass remains constant, and thus the resulting shape is more dense. If m is the constant mass of the pasta in grams and V is the volume of the pasta in mL, then the density p is given by p= - m (a) If the mass remains constant while the volume changes with time, find dp dt (b) Suppose the mass of the pasta is 80g, the density is 0.8 g/mL, and the volume is decreasing at a rate of 400 mL/sec. How fast is the density of the pasta changing at that moment? mL · sec
A tank contains 1,000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 15 L/min. Let y(t) be the amount of salt (in kg) in the tank after t minutes. (a) Find the rate of change of amount of salt in the tank after t minutes. dy dt kg min How much salt is in the tank when t = 0? y(0) How much salt (in kg) is in the tank after t minutes? y = = kg (b) How much salt (in kg) is in the tank after 50 minutes? (Round the answer to one decimal place.) y(50) = kg
Suppose the number of bacteria in a culture increases by 50% every hour if left on its own. Assuming that biologists decide to remove approximately one thousand bacteria from the culture every 10 minutes, which of the following equations best models the population P = P(t) of the bacteria culture, where t' is in hours? A. dP = 5P-1000 dt dP B. at dP C. dt dP D. dt E. dP at = 5P-6000 1.5P-6000 = 1.5P-1000 = -5P-1000
A cylindrical tank holds 22 liters of water. At time t = 0, two taps are opened simultaneously, the upper tap that feeds the water tank with a constant speed of v1 liters per minute and the lower tap that expels the water from the tank with a constant speed of v2 liters per minute . Suppose that v1 = 11 liters / minute and v2 = 3 liters / minute. If the tank has a capacity of 75 liters, when will the tank be filled?
Find the salt pls pounds

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