Problem 46 - Calculus: Early Transcendentals

Question

The air in a room with volume 180 \( m^3 \) contains \( 0.15% \) carbon dioxide initially. Fresher air with only \( 0.05% \) carbon dioxide flows into the room at a rate of 2 \( m^3/min \) and the mixed air flows out at the same rate. Find the percentage of carbon dioxide in the room as a function of time. What happens in the long run?

Step-by-Step Solution

Answer

The percentage of carbon dioxide approaches 0.05% in the long run.

Step 1: Define Initial Conditions

Define the initial conditions given in the exercise. The initial concentration of carbon dioxide in the room is 0.15%. The volume of the room is 180 \( m^3 \). The fresh air entering has a concentration of 0.05% carbon dioxide, and the exchange rate of air is 2 \( m^3/min \).

Step 2: Set Up the Differential Equation

Let \( C(t) \) be the concentration of carbon dioxide in the room at time \( t \). The rate of change of the amount of carbon dioxide in the room is given by the difference between the carbon dioxide entering and leaving. The differential equation is \[ \frac{dC}{dt} = \frac{2}{180}(0.05 - C(t)). \]

Step 3: Solve the Differential Equation

The equation \( \frac{dC}{dt} = \frac{2}{180}(0.05 - C(t)) \) is a first-order linear differential equation. Its solution is of the form \[ C(t) = 0.05 + (C_0 - 0.05)e^{-\frac{2}{180}t}, \] where \( C_0 = 0.15 \) is the initial concentration.

Step 4: Substitute Initial Condition

Substitute \( C_0 = 0.15 \) into the solution to get \[ C(t) = 0.05 + (0.15 - 0.05) e^{-\frac{1}{90}t} = 0.05 + 0.1 e^{-\frac{1}{90}t}. \] This function describes the percentage of carbon dioxide in the room over time.

Step 5: Analyze the Long-Term Behavior

As \( t \to \infty \), \( e^{-\frac{1}{90}t} \to 0 \). Therefore, \( C(t) \to 0.05 \). This indicates that in the long run, the percentage of carbon dioxide in the room approaches 0.05%.

Key Concepts

Carbon Dioxide Concentration

When considering the air quality in a room, one of the major aspects is the concentration of carbon dioxide. Initially in the described room, the carbon dioxide concentration is at 0.15%. Over time, as fresh air flows into the room, the concentration changes, becoming dynamic as it interacts with the introduced air. In the given scenario, an essential part of the analysis is how this concentration changes with time. This change is modeled by the function \( C(t) \), which provides the carbon dioxide percentage at any time \( t \). The differential equation \( \frac{dC}{dt} = \frac{2}{180}(0.05 - C(t)) \), represents this dynamic change. The initial condition, which is the starting percentage before any fresh air enters, is critical for setting our model. With an initial concentration \( C_0 = 0.15 \), the model helps predict how quickly the carbon dioxide levels drop as fresh air replaces the room air.

Air Exchange Rate

Understanding the air exchange rate is crucial when determining how quickly the air in the room changes. In this problem, the rate at which air enters and exits the room is 2 \( m^3/min \). This consistent exchange rate ensures a continual refreshment of the air, affecting the carbon dioxide concentration significantly. The incoming air has a lower carbon dioxide concentration of 0.05%. Hence, the described air exchange forces the concentration in the room to adjust gradually toward this lower value. This process can be mathematically described using a first-order linear differential equation. It captures the balance between what goes into the room and what comes out. Through this steady flow and the differential equation solution, we attain \( C(t) = 0.05 + 0.1 e^{-\frac{1}{90}t} \) which shows how quickly the room adapts to the incoming fresh air.

Long-Term Behavior

An important aspect of the problem is what happens to the carbon dioxide concentration over time, particularly the long-term behavior. As time \( t \) goes to infinity, the term \( e^{-\frac{1}{90}t} \) tends towards zero because of the nature of the exponential decay. This simplification shows that \( C(t) \) approaches 0.05%, the concentration of carbon dioxide in the incoming fresh air. Therefore, the solution reveals that, regardless of the initial concentration, the room will eventually match the carbon dioxide level of the incoming air. This understanding is vital in scenarios like these where air quality regulation is needed, as it ensures that over time the carbon dioxide level will stabilize at a more acceptable and sustainable level of 0.05%.