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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
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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

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).


y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ

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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} Tumor growth The growth of cancer tumors may be modeled by the Gompertz growth equation. Let M(t) be the mass of a tumor, for t ≥ 0. The relevant initial value problem is:


dM/dt = -rM(t)ln(M(t)/K), M(0) = M₀,


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


b. Graph the solution for M₀ = 100 and r = 0.05.

Textbook Question

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 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.

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.

b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.


x′(t) = 2x − 4xy, y′(t) = −y + 2xy

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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−1)(1+y)

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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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