Textbook Question
Growth rate functions
a. Show that the logistic growth rate function f(P)=rP(1−P/K) has a maximum value of rK/4 at the point P=K/2.
Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 9, Problem 9.5.23a
23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.
a. Write an initial value problem for the mass of the substance.
A 500-L tank is initially filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min.
Verified step by step guidance1
Define the variable: Let \(m(t)\) represent the mass of copper sulfate (in grams) in the tank at time \(t\) (in minutes).
Identify the inflow rate of copper sulfate: The solution enters at 4 L/min with a concentration of 20 g/L, so the mass inflow rate is \(4 \times 20 = 80\) g/min.
Identify the outflow rate of copper sulfate: Since the tank is well mixed, the concentration inside the tank at time \(t\) is \(\frac{m(t)}{500}\) g/L. The outflow rate is 4 L/min, so the mass outflow rate is \(4 \times \frac{m(t)}{500} = \frac{4m(t)}{500}\) g/min.
Write the differential equation expressing the rate of change of mass in the tank: \(\frac{dm}{dt} = \) (mass inflow rate) \(-\) (mass outflow rate), which gives \(\frac{dm}{dt} = 80 - \frac{4m(t)}{500}\).
Specify the initial condition: Since the tank is initially filled with pure water, the initial mass of copper sulfate is zero, so \(m(0) = 0\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Formulating Initial Value Problems (IVPs)
An initial value problem involves setting up a differential equation that models the rate of change of a quantity along with an initial condition. In this context, it means expressing how the mass of copper sulfate in the tank changes over time, starting from zero since the tank initially contains pure water.
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Initial Value Problems
Mass Balance in Continuous Stirred Tank Reactors (CSTR)
A mass balance accounts for the mass entering, leaving, and accumulating in the tank. For a stirred tank with inflow and outflow at equal rates, the change in mass equals the mass entering minus the mass leaving, assuming perfect mixing ensures uniform concentration throughout the tank.
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03:38Intro to Continuity Example 1
Differential Equations for Concentration and Volume
The problem involves setting up a first-order linear differential equation relating the mass of solute to time, considering constant volume due to equal inflow and outflow rates. Concentration is mass divided by volume, and the inflow concentration and flow rate determine the input term in the equation.
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Related Practice
Textbook Question
{Use of Tech} Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation
dP/dt=kP(1−P/A),P0=P_0,
where K is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t=0. The model also assumes no recovery.
a. Find the solution of the initial value problem, for t≥0, in terms of K, A, and P0.
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Textbook Question
U.S. population projections According to the U.S. Census Bureau, the nation’s population (to the nearest million) was 296 million in 2005 and 321 million in 2015. The Bureau also projects a 2050 population of 398 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Here is one approach:
a. Assume t = 0 corresponds to 2005 and that the population growth is exponential for the first ten years; that is, between 2005 and 2015, the population is given by P(t) = P(0)exp(rt). Estimate the growth rate r using this assumption.
Textbook Question
A second-order equation Consider the differential equation y''(t) - k²y(t) = 0 where k > 0 is a real number.
a. Verify by substitution that when k = 1, a solution of the equation is y(t) = C₁eᵗ + C₂e⁻ᵗ. You may assume this function is the general solution.
Textbook Question
A physiological model A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time t = 0, an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of 500 mg/L. The inflow rate is 0.06 L/min. Assume the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant.
a. Write an initial value problem that models the mass of the drug in the blood, for t ≥ 0.
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Textbook Question
Solving Bernoulli equations Use the method outlined in Exercise 43 to solve the following Bernoulli equations.
a. y′(t) + y = 2y²
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