91Ó°ÊÓ

Carbon monoxide (CO) is a colorless, odorless gas, and when indoors, its concentration levels are crucial for safety. In our exercise, the initial CO concentration is given as 3% inside the room. This is a dangerously high level. For safe air quality, it must be reduced to 0.01%, a much safer level.

The concentration of carbon monoxide in a room depends on various factors such as the size of the room, the rate at which fresh air enters and exits the room, and the initial concentration of CO. For this reason, understanding how concentration changes over time with ventilation is important in this problem.

In this scenario, fresh air, devoid of carbon monoxide, is introduced into the room to dilute the CO concentration. At the same time, the room is vented so that the air mixture, now containing CO, leaves at an equal rate, effectively reducing the CO concentration.

Exponential decay is a process whereby the quantity decreases at a rate proportional to its current value. This principle is vividly displayed in our exercise, where the amount of carbon monoxide in the room follows an exponential decay pattern.

The mathematical form for exponential decay is expressed as:\[ A(t) = A(0)e^{-kt} \]where:- \( A(t) \) is the amount of substance at time \( t \),- \( A(0) \) is the initial amount,- \( k \) is the decay constant,- \( e \) is the base of the natural logarithm.

In the case of the room problem:

• \( A(0) = 0.03 \times V \) represents the initial carbon monoxide in the room.
• The decay constant \( k = \frac{Q}{V} \), which regulates how quickly the CO concentration reduces over time.

Exponential decay provides us with a predictable pattern by which we can calculate when the CO concentration has decayed to a desired lower level such as 0.01%.

A first order linear equation is a differential equation that relates a function and its first derivative. These are often of the form:\[ \frac{dy}{dt} + P(t)y = Q(t) \]Such equations are often found in real-world problem situations, like the carbon monoxide concentration problem, to describe rates of change.

In our exercise, the differential equation is given as:\[ \frac{dA}{dt} = -\frac{A(t)}{V}Q \]where:- \( A(t) \) is the amount of carbon monoxide at time \( t \).- \( V \) is the volume of the room.- \( Q \) is the flow rate of the air.

This differential equation exemplifies a first order linear equation, showing how changes in CO concentration are directly related to their current levels. Solving this type of equation, usually by methods like integrating factors, allows us to find solutions that describe how the system evolves over time.