Skip to main content
Ch. 9 - First-Order Differential Equations
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 9 - First-Order Differential Equations문제 9.3.16
9장, 문제 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%.

검증된 단계별 안내
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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
10m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
07:39
Classifying Differential Equations

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.
추천 영상:
3:00
Determining Different Coordinates for the Same Point Example 2

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.
추천 영상:
06:06
Solving Separable Differential Equations
관련 실천
교과서 질문

Use Euler’s method with dx = 0.2 to estimate y(2) if y′ = y/x and y(1) = 2. What is the exact value of y(2)?

31
views
교과서 질문

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


tan θ dr/dθ + r = sin²θ, 0 < θ < π/2

17
views
교과서 질문

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

38
views
교과서 질문

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

22
views
교과서 질문

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' = 2y²(x-1), y(2) = -1/2, dx = 0.1, x* = 3

24
views
교과서 질문

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)?

27
views