Answered: 3. Solve the following initial value… | bartleby

College Algebra

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Systems Of Equations And Inequalities

Section6.5: Systems Of Inequalities

Problem 9ECP

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Answer question 3 please

1. Consider two interconnected tanks. Tank 1 initially contains 30 gallons of water and 25 oz. of salt, and Tank 2
initially contains 20 gallons of water and 15 oz. of salt. Water containing 1 oz/gal of salt flows into Tank 1 at a
rate of 1.5 gal/min. The mixture flows from Tank 1 to Tank 2 at a rate of 3 gal/min. Water containing 3 oz/gal
of salt also flows into Tank 2 at a rate of 1 gal/min (from the outside). The mixture drains from Tank 2 at a
rate of 4 gal/min, of which some flows back into Tank 1 at a rate of 1.5 gal/min, while the remainder leaves
the system.
(a) Let S1(t) and S2(t), respectively, denote the amount of salt in each tank at time t. Write down the
differential equations and initial conditions that model this flow process.
(b) Find the values of S1 and S2 at which the system is in equilibrium; represent these values by S; and S
respectively. Which tank do you think will approach its equilibrium value most rapidly?
(c) Introduce the new variables x1 = S1-S and x2 = S2-S, to represent the the deviation from equilibrium
values. Rewrite the system in terms of these new variables.
(d) Solve the system for x1 and x2 and graph the time-series plots for x1 and x2 (use the same set of axes).
(e) Estimate the time T at which |x1(t)|<0.5 and |x2(t)| < 0.5 for all t 2T.
2. A Swaying Skyscraper. When the John Hancock Building in Boston, MA was first built it tended to
and forth (like an oscillating spring), so much so that people in the top floors experienced motion sickness.
Similar to the spring-mass system of differential equations, we can model the swaying of the building by
adding a gravity term to the spring mass model; this term turns out to be x. Notice how this system is
similar to a linear system but has the addition of a nonlinear term x in the y' equation. The inclusion of
this nonlinear term means that we are not able to use the straight line solutions or eigenvalue techniques.
However, we can still use our previous graphical approaches to explore solutions to help make predictions
about the swaying motion of the building.
sway
back
dx
= y
dt
dy
= -x- y+ x³
dt
In this simplified system of rate of change equations, x measures the amount of displacement of the building
from the vertical position at any time t and y measures the horizontal velocity of the building at any time t.
Use PPLANE or the GeoGebra Vector Field applet, https://ggbm.at/kkNXUVds, as a tool to explore solutions
as viewed in the phase plane.
(a) Determine all equilibrium solutions and explain the meaning of each one in terms of the swaying skyscraper.
(b) Provide a sketch of several representative curves in the phase plane and give an interpretation for the
motion of the building for the different types of curves (e.g., does the building remain standing? If so, for
what initial conditions? For what range of initial conditions is a disaster predicted?)
3. Solve the following initial value problems using Laplace transforms.
(a) y' + 5y= sin(2t), y(0) = 0.
%3D
(b) у"+2у'+у-е", у(0) %3 0,
y'(0) = 0.

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20.Consider two interconnected tanks similar to those in Figure 7.1.6. Initially, Tank 1 contains 60 gal of water and Q01Q10 oz of salt, and Tank 2 contains 100 gal of water and Q02Q20 oz of salt. Water containing q1 oz/gal of salt flows into Tank 1 at a rate of 3 gal/min. The mixture in Tank 1 flows out at a rate of 4 gal/min, of which half flows into Tank 2, while the remainder leaves the system. Water containing q2 oz/gal of salt also flows into Tank 2 from the outside at the rate of 1 gal/min. The mixture in Tank 2 leaves it at a rate of 3 gal/min, of which some flows back into Tank 1 at a rate of 1 gal/min, while the rest leaves the system. a.Draw a diagram that depicts the flow process described above. Let Q1(t) and Q2(t), respectively, be the amount of salt in each tank at time t. Write down differential equations and initial conditions for Q1 and Q2 that model the flow process. b.Find the equilibrium values QE1Q1E and QE2Q2E in terms of the concentrations q1 and q2. c.Is it…
Consider the following tank-water system. Tank A holds 200 liters of water, initially pure. Brine with a concentration of 3 grams of salt per liter flows into tank A at 5 liters per minute. The solution flows out of tank A into tank B at a rate of 5 liters per minute. Tank B is a 125 liter tank of initially pure water which receives the outflow from tank A, and the well-mixed solution flows out of tank B at 5 liters per minute. Write down a system of differential equations that models the scenario.
solve and draw a diagram of the problem Two large tanks, each holding 100L of liquid, are interconnected by pipes, with the liquid flowing from tank A into tank B at a rate of 3L/min and from tank B into tank A at a rate of 1L/min. The liquid inside each tank is kept well stirred. Pure water flows into tank A at a rate of 6L/min. The solution flows out of the system from tank A at 4L/min and from tank B at 2L/min. If, initially, tank A contains pure water and tank B contains 20kg of salt, determine the mass of salt in each tank at time t ≥ 0.
Consider the two-tank connection pictured below. The salt water flows at specified rates and thesolution is assumed to be “well-stirred”. Each tank has a capacity of 5000 gal. Initially, Tank 1 has 500 gal. of fresh water and Tank 2 has 800 gal. of salt water with 2 lb of salt dissolved in it. Salt water with a concentration of 1/5 lb/gal enters Tank 1 at a rate of 4 gal/min and the mixture leaves at a rate of 7 gal/min at the same time. Salt water with a concentration of 1/4 lb/gal enters Tank 2 at a rate of 10 gal/min and the mixture leaves at a rate of 9 gal/min at the same time. Additionally, the mixture from Tank 1 flows into Tank 2 at a rate of 3 gal/min andthe mixture from Tank 2 flows into Tank 1 at a rate of 8 gal/min. Let Q1(t) and Q2(t) denote theamounts of salt in pounds in Tanks 1 and 2 at time t in minutes. Set up the initial value problemsfor Q1(t) and Q2(t) at time t. (Do not solve.)
Two connected tanks, each with a capacity of 50 liters, contain brine (saltwater). Initially, the first tank contains 18 liters of brine with a salt concentration of 3 grams per liter and the second contains 17 liters of brine with a salt concentration of 2 grams per liter. At t = 0 brine with a salt concentration of 6 grams per liter flows into the first tank at 8 liters per hour. Well-stirred brine flows from the first tank into the second at 7 liters per hour, from the second into the first at 5 liters per hour, from the first into a drain at 4 liters per hour, and from the second into a drain at 3 liters per hour. (a) Determine the volume (liters) of brine in each tank as a function of time.(b) Give an initial-value problem that governs the amount (grams) of salt in eachtank as a function of time.(c) Give the interval of definition for the solution of this initial-value problem.
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A refinery manufactures two grades of jet fuel, Fl and F2, by blending four types of gasoline, A. B, C, and D. Fuel Fl uses gasolines A. B. C, and D in the ratio 1:1:2:4, and fucl F2 uses the ratio 2:2:1:3. The supply limits for A, B.C, and D are 1000, 1200, 900, and 1500 bbl/day, respectively. The costs per bbl for gasolines A, B, C, and D are $120, $90, $100, and $150, respectively. Fucls Fl and F2 sell for $200 and $250 per bbl, respectively. The minimum demand for F1 and F2 is 200 and 400 bbl/day, respectively. Develop an LP model to determine the optimal production mix for F1 and F2, and find the solution using execl Solver
A chicken farmer can buy a special food mix A at 20c per Kg and special food mix B at 40c perKg. Each Kg of mix A contains 3000 units of nutrient N1 and 1000 units of nutrient N2; eachKg of mix B contains 4000 units of nutrient N1 and 4000 units of nutrient N2. If the minimumdaily requirements for the chickens collectively are 36000 units of nutrient N1 and 20000 unitsof nutrient N2, how many pounds of each food mix should be used each day to minimize dailyfood costs while meeting (or exceeding) the minimum daily nutrient requirements? What is theminimum daily cost?
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