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A lecture theatre having volume \(1000 \mathrm{~m}^{3}\) is designed to seat 200 people. The air is conditioned continuously by an inflow of fresh air at a constant rate \(V\left(\right.\) in \(\mathrm{m}^{3} \mathrm{~min}^{-1}\) ). An average person generates \(980 \mathrm{~cm}^{3}\) of \(\mathrm{CO}_{2}\) per minute, while fresh air contains \(0.04 \%\) of \(\mathrm{CO}_{2}\) by volume. Show that the percentage concentration \(x\) of \(\mathrm{CO}_{2}\) by volume in the lecture theatre at time \(t\) (in min) after the audience enters satisfies the differential equation $$ 1000 \frac{\mathrm{d} x}{\mathrm{~d} t}=19.6+0.04 V-V x(t) $$ If initially \(x(0)=0.04\), show that \(x(t)\) is an increasing function of \(t\) for \(t>0\). Deduce that the maximum \(x^{*}\) of \(x(t)\) is given by $$ x^{8}=(19.6+0.04 \mathrm{~V}) / \mathrm{V} $$ If the specification is that \(x\) does not exceed \(0.06\) (that is, \(50 \%\) increase above fresh air), deduce that \(V\) must be chosen so that \(V>980 .\) Comment on this result.

Step by step solution

01

Understand the Problem

We need to analyze the flow of carbon dioxide (CO₂) in a lecture theatre with a volume of 1000 m³ containing 200 people. Each person generates 980 cm³ of CO₂ per minute. Simultaneously, fresh air with a CO₂ concentration of 0.04% is flowing in at a constant rate, \( V \) m³/min. We are to derive and solve the differential equation governing the CO₂ concentration over time.

02

Set Up the Differential Equation

We set the differential equation by analyzing the rates of CO₂ inflow and outflow. The inflow of CO₂ due to people is \( 200 \times 980 \) cm³/min, which is equivalent to 19.6 m³/min. The inflow due to fresh air is \( 0.0004V \) m³/min. The outflow is expressed as \( Vx(t) \), where \( x(t) \) is the CO₂ concentration in m³/m³. Thus, the differential equation is \( 1000 \frac{dx}{dt} = 19.6 + 0.04V - Vx(t) \).

03

Check Initial Condition and Behavior

Initially, \( x(0) = 0.04 \). We need to show \( x(t) \) is increasing. We can consider the differential equation: \( \frac{dx}{dt} = \frac{19.6 + 0.04V - Vx(t)}{1000} \). Since \( Vx(t) > 0 \), as long as \( 19.6 + 0.04V > 0 \), \( \frac{dx}{dt} > 0 \), implying \( x(t) \) is increasing.

04

Find Condition for Maximum Value

To find the maximum value \( x^* \), we set \( \frac{dx}{dt} = 0 \), giving \( 19.6 + 0.04V - Vx^* = 0 \). Solving for \( x^* \), we get \( x^* = \frac{19.6 + 0.04V}{V} \).

05

Determine Condition on Inflow Rate \( V \)

Given the condition \( x^* \leq 0.06 \), substitute \( x^* = \frac{19.6 + 0.04V}{V} \) and rearrange to find \( V > 980 \). This ensures \( x(t) \) does not exceed 0.06, maintaining COâ‚‚ concentration within acceptable levels.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Carbon Dioxide Concentration

Carbon dioxide (COâ‚‚) concentration refers to the amount of COâ‚‚ present in the air, usually expressed as a percentage or a fraction of the total volume of air. In this particular problem, COâ‚‚ concentration is measured in a lecture theatre where the air needs to be managed carefully to ensure a safe and comfortable environment for the occupants.

COâ‚‚ in the lecture theatre comes from two primary sources:

• Human respiration: As individuals exhale, they add to the COâ‚‚ levels. In our problem, each person contributes 980 cm³ of COâ‚‚ per minute.
• Fresh air inflow: Although fresh air is considered clean, it still contains a small percentage (0.04%) of COâ‚‚.

By understanding these contributions, we can model the overall COâ‚‚ concentration in the room and make necessary adjustments through ventilation.

Air Conditioning Systems

Air conditioning systems play a critical role in controlling the indoor air quality by regulating the temperature, humidity, and most importantly, the concentration of gases like carbon dioxide. In the problem setting of the lecture theatre, the air conditioning system is responsible for circulating fresh air into the room at a constant rate, denoted by the variable \( V \) in cubic metres per minute.

This system ensures that stale air is consistently being replaced with fresh air, which helps in:

• Maintaining optimal COâ‚‚ concentration levels.
• Enhancing comfort for the occupants.
• Ensuring health and safety standards are met by preventing excessive COâ‚‚ build-up.

Proper ventilation is crucial in high occupancy areas as it dilutes the COâ‚‚ being generated by people within the space and prevents the air from becoming stuffy or hazardous.

Rate of Change

In mathematics, the rate of change refers to how quickly a quantity changes over time. In the context of this problem, we're discussing the rate of change in the concentration of COâ‚‚ in the lecture theatre. This concept is integral since it helps us understand whether the COâ‚‚ concentration is increasing or decreasing at any given time after the audience has entered the theatre.

• The rate of change is determined by the differential equation \( \frac{dx}{dt} = \frac{19.6 + 0.04V - Vx(t)}{1000} \).
• This equation is crucial because it allows us to predict the COâ‚‚ level at various times and helps in assessing if the air conditioning system is adequate.

This function is essential for ensuring that the rate of change remains positive (meaning COâ‚‚ levels increase) only within acceptable bounds, thereby helping maintain air quality standards in indoor environments.

Maximum Concentration

Determining the maximum concentration of COâ‚‚ is vital for ensuring safety and comfort in indoor spaces such as the lecture theatre. The maximum concentration, denoted as \( x^* \), represents the highest level of COâ‚‚ that the room will reach under the given conditions of ventilation and occupancy.

The formula derived from setting the rate of change to zero is \( x^* = \frac{19.6 + 0.04V}{V} \). This equation provides a ceiling for how high the concentration can rise, based on the inflow rate \( V \).

To adhere to safety standards, which dictate that \( x^* \) should not exceed 0.06, the equation helps us calculate the minimum ventilation rate needed to keep the COâ‚‚ concentration at this safe level. Therefore, the problem concludes that \( V \) must be greater than 980 cubic metres per minute to meet this requirement, ensuring a balance between sufficient airflow and manageable COâ‚‚ levels.