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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.5.26b
Chapter 9, Problem 9.5.26b

23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.
b. Solve the initial value problem.


A one-million-liter pond is contaminated by a chemical pollutant with a concentration of 20 g/L. The source of the pollutant is removed, and pure water is allowed to flow into the pond at a rate of 1200 L/hr. Assuming the pond is thoroughly mixed and drained at a rate of 1200 L/hr, how long does it take to reduce the concentration of the solution in the pond to 10% of the initial value?

Verified step by step guidance
1
Define the variables: let \(C(t)\) be the concentration of the pollutant in the pond at time \(t\) (in hours), measured in grams per liter (g/L). The initial concentration is \(C(0) = 20\) g/L.
Set up the differential equation based on the mixing and flow rates. Since pure water flows in and the mixture flows out at the same rate, the volume remains constant at 1,000,000 L. The rate of change of pollutant mass in the pond is given by the difference between pollutant inflow and outflow:
\[ \frac{d}{dt} [\text{mass of pollutant}] = \text{inflow rate} - \text{outflow rate} \]
Since the inflow is pure water, the inflow rate of pollutant is zero. The outflow rate of pollutant is the concentration times the outflow volume rate:
\[ \frac{d}{dt} (V \cdot C) = 0 - (1200 \text{ L/hr}) \times C(t) \]
Because the volume \(V\) is constant, rewrite the equation as:
\[ V \frac{dC}{dt} = -1200 C(t) \]
Divide both sides by \(V\) to get the first-order linear differential equation:
\[ \frac{dC}{dt} = -\frac{1200}{1,000,000} C(t) \]
Solve this differential equation using separation of variables or recognizing it as an exponential decay:
\[ C(t) = C(0) e^{-\frac{1200}{1,000,000} t} \]
Use the condition that the concentration reduces to 10% of the initial value, i.e., \(C(t) = 0.1 \times 20 = 2\) g/L, and solve for \(t\):
\[ 2 = 20 e^{-\frac{1200}{1,000,000} t} \]
Take the natural logarithm of both sides and solve for \(t\) to find the time required to reach the desired concentration.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Formulating the Differential Equation for Mixing Problems

In stirred tank problems, the concentration changes over time due to inflow and outflow. The rate of change of pollutant mass is modeled by a first-order differential equation, balancing the pollutant entering and leaving the system. Since pure water flows in, only outflow reduces concentration, leading to a separable ODE describing concentration decay.
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Classifying Differential Equations

Solving Initial Value Problems (IVPs)

An initial value problem involves solving a differential equation with a given initial condition. Here, the initial concentration is known, and the solution describes concentration over time. Techniques like separation of variables or integrating factors help find explicit formulas to predict pollutant levels at any time.
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Initial Value Problems

Exponential Decay and Time to Reach a Specific Concentration

When a pollutant concentration decreases proportionally to its current value, the solution exhibits exponential decay. To find the time to reach a certain fraction of the initial concentration, set the solution equal to that fraction and solve for time, often involving natural logarithms.
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Related Practice
Textbook Question

27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.


c. Find the equilibrium points for the system.


x′(t) = −3x + 6xy, y′(t) = y − 4xy

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Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

b. Sketch the direction field, for t≥0.


y′(t) = 2y + 4

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Textbook Question

{Use of Tech} Tumor growth The Gompertz growth equation is often used to model the growth of tumors. Let M(t) be the mass of a tumor at time t≥0. The relevant initial value problem is 

dM/dt=−rM ln(M/K),M(0)=M0, 

where r and K are positive constants and 0<M0<K.

b. Solve the initial value problem and graph the solution for r=1,K=4, and M0=1. Describe the growth pattern of the tumor. Is the growth unbounded? If not, what is the limiting size of the tumor? 

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Textbook Question

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.


b. In what regions are solutions increasing? Decreasing?


y'(t) = y(y+3)(4-y)

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Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

b. Sketch the direction field, for t≥0. 


y′(t) = 6 - 2y

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Textbook Question

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.


{Use of Tech} Free fall One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation v'(t) = g - bv, where v(t) is the velocity of the object for t ≥ 0, g = 9.8 m/s² is the acceleration due to gravity, and b > 0 is a constant that involves the mass of the object and the air resistance.


c. Using the graph in part (b), estimate the terminal velocity lim(t→∞) v(t).

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