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- Two very large tanks A and B are each partially filled with 100 gallons of brine. Initially, 100 pounds of salt is dissolved in the solution in tank A and 50 pounds of salt is dissolved in the solution in tank B. The system is closed in that the well-stirred liquid is pumped only between the tanks, as shown in Figure 3.3.7.(a) Use the information given in the figure to construct a mathematical model for the number of pounds of salt x1 (t) and x2(t) at time t in tanks A and B, respectively.(b) Find a relationship between the variables x1(t) and x2 (t) that holds at time t. Explain why this relationship makes intuitive sense. Use this relationship to help find the amount of salt in tank B at t = 30 min. FIGURE 3.3.7 Mixing tanks in Problem 9Chapter 3.3, Problem 9E, Two very large tanks A and B are each partially filled with 100 gallons of brine. Initially, 100Use the Gauss-Seidel method without relaxation to solve the following system of equations to a tolerance of Es=5% (percent relative error). If necessary, rearrange the equations to achieve convergence. Start with [X]T= [1.5, 2.5, 4.5]6x1 - x2 - x3 = 3-3x1 + x2 + 12x3 = 506x1 + 9x2 + x3 = 40I only need question 4, Thank you. In the presence of enough food and lacking predators and competitors, a population of rabbitswill increase by a fixed percentage each spring. For this set of exercises, we will say that therabbit population increases by 10% each year. Thus, if the initial population is R0, then thepopulation the following spring will be R1 = (1.1)R0 rabbits. In two years, the population will beR2 = (1.1)R1 = (1.1)2R0. The number of rabbits after n years will beRn = (1.1)nR0,and clearly the rabbit population grows without bound. We will assume that the rabbit populationis measured in hundreds of rabbits (so that R = 1 represents 100 rabbits).Suppose now that we introduce a small number of cougars into the environment to keep the rabbitpopulation under control.Let’s suppose that the amount of rabbits eaten each year is proportional to the cougar population.Then the change in the rabbit population is governed by the equationRn+1 = (1.1)Rn − (0.1)Cn,where Rn and Cn…
- Let us consider an aging spring - mass system where the restoring force of the spring and the damping force are both weakening exponentially over time. Let the equation of motion of the mass be governed by the following initial value problem: 2?″(?)+2?−0.1??′(?)+4?−0.2??(?)=0,?(0)=1,?′(0)=0 Using what you have learned in this module, find the first four nonzero terms in the series solution of the above IVP about ?=0. You may use the following facts: ???=∑∞?=0(??)??! (Standard Taylor Series of ??? about ?=0) (∑∞?=0????)(∑∞?=0????)=∑∞?=0???? , where ??=?0??+?1??−1+⋅⋅⋅+???0 (Rule for multiplying two series)Illustrate the Picard iteration scheme for the initial value problem y' = x - y, y(0) = 1.Two very large tanks A and B are partially filled with 100 gallons of brine each. Initially, 100 pounds of salt are dissolved in the solution in tank A and 50 pounds of salt are dissolved in the solution in tank B. The system is closed, since the well-mixed liquid is pumped only between the tanks as shown in the figure. 1. Use the information in the figure to construct a mathematical model for the number of pounds of salt x1(t) and x2(t) at time "t" in tanks A and B, respectively. 2. Find a relationship between the variables x1(t) and x2(t) that holds at time «t». 3. Explain why this relationship makes intuitive sense. 4. Use this relationship to help find the amount of salt in tank B at t = 30 min.
- Let us given the following system7x + 2y − 3z = 17 3x − 6y − z = −8 3x − 2y − 6z = −2. (a) Write the Gauss-Seidel iteration sequence for the given system. (b) Does the the Gauss-Seidel iteration in part (a) converge for any choice of the initial point (x0, y0, z0)? Explain.I only need question 3 answered, thank you. In the presence of enough food and lacking predators and competitors, a population of rabbitswill increase by a fixed percentage each spring. For this set of exercises, we will say that therabbit population increases by 10% each year. Thus, if the initial population is R0, then thepopulation the following spring will be R1 = (1.1)R0 rabbits. In two years, the population will beR2 = (1.1)R1 = (1.1)2R0. The number of rabbits after n years will beRn = (1.1)nR0,and clearly the rabbit population grows without bound. We will assume that the rabbit populationis measured in hundreds of rabbits (so that R = 1 represents 100 rabbits).Suppose now that we introduce a small number of cougars into the environment to keep the rabbitpopulation under control.Let’s suppose that the amount of rabbits eaten each year is proportional to the cougar population.Then the change in the rabbit population is governed by the equationRn+1 = (1.1)Rn − (0.1)Cn,where Rn and…Consider the following system. x+5y=9 4x-y=15 Start with P0=0 and use Gauss-Seidel Iteration to find Pk (k=1,2). Options for the value of x1, y1, x2, y2 include: 3.75 1.05 9 21 1.0002 4.0005 0.9975 -96 -399 Will Gauss-Seidel Iteration converge to the solution? A) Yes B) No
- Two interacting populations of hares and foxes can be modeled by the recursive equations h(t + 1) = 4h(t) − 2 f (t) f(t+1)=h(t)+ f(t). For each of the initial populations given in parts (a) through (c), find closed formulas for h(t) and f (t). a. h(0)= f(0)=100 b. h(0)=200, f(0)=100 c. h(0)=600, f(0)=500Find the solution of the following initial-value problemFor the system below, analyze the convergence of the iterative Gauss-Jacobi method and apply this method (swap lines, if necessary) to find an approximation of the solution of thesystem, with ε = 10−2