The air in a room with volume 180 m³ contains 0.3% carbon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of 2 m³/min and the mixed air flows out at the same rate. Find the percentage p of carbon dioxide in the room as a function of time t (in minutes). p(t) = 0.5556e % +0.7 X What happens to the percentage of carbon dioxide in the room in the long run? lim p(t) = 0.007 X %
The air in a room with volume 180 m³ contains 0.3% carbon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of 2 m³/min and the mixed air flows out at the same rate. Find the percentage p of carbon dioxide in the room as a function of time t (in minutes). p(t) = 0.5556e % +0.7 X What happens to the percentage of carbon dioxide in the room in the long run? lim p(t) = 0.007 X %
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Topic: Carbon Dioxide Concentration in a Room Over Time
#### Problem Statement
The air in a room with a volume of \(180 \, \text{m}^3\) contains \(0.3\%\) carbon dioxide initially. Fresher air with only \(0.05\%\) carbon dioxide flows into the room at a rate of \(2 \, \text{m}^3/\text{min}\), and the mixed air flows out at the same rate.
#### Task 1: Function of Carbon Dioxide Percentage Over Time
Find the percentage \( p \) of carbon dioxide in the room as a function of time \( t \) (in minutes).
\[ p(t) = 0.5556e^{\left( \frac{-1}{90} t \right)} + 0.7 \]
Here we have a mathematical function describing the decay of carbon dioxide concentration over time.
#### Task 2: Long-Term Behavior of Carbon Dioxide Percentage
What happens to the percentage of carbon dioxide in the room in the long run?
\[ \lim_{t \to \infty} p(t) = 0.007 \]
The limit indicates the long-term behavior of the carbon dioxide concentration as time approaches infinity, showing that it will stabilize at \(0.007\%\).
### Explanation of Graphs/Diagrams
In this scenario, there are no graphical representations or diagrams provided. However, here's an explanation of the text and the formula within the context:
- \[ p(t) \] represents the percentage of carbon dioxide in the room at any given time \( t \).
- The form of the function \(0.5556e^{\left( \frac{-1}{90} t \right)} + 0.7\) suggests an exponential decay of the carbon dioxide concentration with a constant term indicating the equilibrium concentration that the room air will approach over time.
##### Interpretation of the Formula:
- \( e^{\left( \frac{-1}{90} t \right)} \) signifies the exponential decay component, where the concentration decreases over time.
- \( 0.5556 \) is a constant multiplier for the exponential term.
- \( 0.7 \) represents the baseline percentage of carbon dioxide that the room will eventually settle at, considering the constant inflow of fresher air with \(0.05\%\) carbon dioxide.
##### Long-Term Analysis:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a1305ef-606e-42df-8570-9f3784bb6037%2Fbfd89520-d307-4dbf-95dc-a14d22840c74%2Fh45epdu_processed.png&w=3840&q=75)
Transcribed Image Text:### Topic: Carbon Dioxide Concentration in a Room Over Time
#### Problem Statement
The air in a room with a volume of \(180 \, \text{m}^3\) contains \(0.3\%\) carbon dioxide initially. Fresher air with only \(0.05\%\) carbon dioxide flows into the room at a rate of \(2 \, \text{m}^3/\text{min}\), and the mixed air flows out at the same rate.
#### Task 1: Function of Carbon Dioxide Percentage Over Time
Find the percentage \( p \) of carbon dioxide in the room as a function of time \( t \) (in minutes).
\[ p(t) = 0.5556e^{\left( \frac{-1}{90} t \right)} + 0.7 \]
Here we have a mathematical function describing the decay of carbon dioxide concentration over time.
#### Task 2: Long-Term Behavior of Carbon Dioxide Percentage
What happens to the percentage of carbon dioxide in the room in the long run?
\[ \lim_{t \to \infty} p(t) = 0.007 \]
The limit indicates the long-term behavior of the carbon dioxide concentration as time approaches infinity, showing that it will stabilize at \(0.007\%\).
### Explanation of Graphs/Diagrams
In this scenario, there are no graphical representations or diagrams provided. However, here's an explanation of the text and the formula within the context:
- \[ p(t) \] represents the percentage of carbon dioxide in the room at any given time \( t \).
- The form of the function \(0.5556e^{\left( \frac{-1}{90} t \right)} + 0.7\) suggests an exponential decay of the carbon dioxide concentration with a constant term indicating the equilibrium concentration that the room air will approach over time.
##### Interpretation of the Formula:
- \( e^{\left( \frac{-1}{90} t \right)} \) signifies the exponential decay component, where the concentration decreases over time.
- \( 0.5556 \) is a constant multiplier for the exponential term.
- \( 0.7 \) represents the baseline percentage of carbon dioxide that the room will eventually settle at, considering the constant inflow of fresher air with \(0.05\%\) carbon dioxide.
##### Long-Term Analysis:
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