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Ch. 9 - First-Order Differential Equations
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 9 - First-Order Differential EquationsProblem 9.3.16
Chapter 9, Problem 9.3.16

Carbon monoxide pollution An executive conference room of a corporation contains 4500 ft³ of air initially free of carbon monoxide. Starting at time t = 0, cigarette smoke containing 4% carbon monoxide is blown into the room at the rate of 0.3 ft³/min. A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of 0.3 ft³/min. Find the time when the concentration of carbon monoxide in the room reaches 0.01%.

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1
Define the variable \(Q(t)\) as the volume of carbon monoxide (in cubic feet) in the room at time \(t\) minutes. Initially, since the room is free of carbon monoxide, we have \(Q(0) = 0\).
Set up the rate of change of carbon monoxide in the room. The rate in is the concentration of carbon monoxide in the incoming air multiplied by the inflow rate: \(0.04 \times 0.3\) ft³/min. The rate out is the concentration of carbon monoxide in the room multiplied by the outflow rate: \(\frac{Q(t)}{4500} \times 0.3\) ft³/min.
Write the differential equation expressing the rate of change of \(Q(t)\) as the difference between the inflow and outflow rates: \(\displaystyle \frac{dQ}{dt} = 0.04 \times 0.3 - \frac{Q(t)}{4500} \times 0.3\).
Simplify the differential equation and solve it as a first-order linear ordinary differential equation. Use an integrating factor or recognize it as a separable equation to find the general solution for \(Q(t)\).
Use the condition that the concentration reaches 0.01%, which means \(\frac{Q(t)}{4500} = 0.0001\), to solve for the time \(t\) when this occurs.

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Key Concepts

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

Modeling with Differential Equations

This problem involves setting up a differential equation to model the changing concentration of carbon monoxide in the room over time. The rate of change depends on the inflow of polluted air and the outflow of mixed air, requiring an understanding of how to translate physical rates into a mathematical equation.
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Mixing and Concentration in a Well-Mixed System

The assumption that the air is well mixed means the concentration of carbon monoxide is uniform throughout the room at any time. This allows the use of a single variable to represent concentration, simplifying the problem and enabling the use of inflow and outflow rates to determine concentration changes.
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Solving First-Order Linear Differential Equations

The resulting differential equation is first-order linear and can be solved using integrating factors or separation of variables. Understanding how to solve such equations is essential to find the concentration as a function of time and determine when it reaches the specified threshold.
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Related Practice
Textbook Question

Integral Equations

In Exercises 7–12, write an equivalent first-order differential equation

and initial condition for y.


y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt

Textbook Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


tan θ dr/dθ + r = sin²θ, 0 < θ < π/2

Textbook Question

In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.


y′ = √x/y, y > 0, y(0) = 1, dx = 0.1, x* = 1

Textbook Question

In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.


y' = 2xexp(x²) , y(0) = 2, dx = 0.1, x* = 1

Textbook Question

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.


y' = x(1-y), y(1) = 0, dx = 0.2

Textbook Question

Use Euler’s method with dx = 1/3 to estimate y(2) if y′ = x sin y and y(0) = 1. What is the exact value of y(2)?