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Imagine a drug that needs to remain effective within your body. For it to work, it must stay above a specific concentration level. In this case, we're looking at a drug with an initial concentration of 640 mg/L that needs to remain effective above 100 mg/L. Understanding how this drug's concentration changes over time in the bloodstream is key. This involves tracking how the concentration dwindles or decays.
One critical factor is that the drug experiences a decay of 20% each hour. This means every hour, the concentration isn't just dropping by a fixed amount - it's decreasing by a percentage of whatever amount is present at that time. This kind of decay is what we refer to as exponential decay.
By creating a model of this decay, we can predict when the drug's concentration will fall below the needed level. This predictive ability is vital, especially in medicine, to ensure that drugs are administered at the right intervals.

To mathematically capture how the drug declines over time, we use the exponential decay formula. This formula is a powerful tool, ideal for processes where a quantity diminishes at a rate proportional to its current value. The general model is denoted as:

• \( C(t) = C_0 (1 - r)^t \)

Here's what each part means:

• \( C(t) \): The concentration at time \( t \)
• \( C_0 \): The initial concentration (here, 640 mg/L)
• \( r \): The decay rate (in our problem, 0.2 or 20%)
• \( t \): Time in hours

Substituting the known values into the formula, we get \( C(t) = 640(0.8)^t \). This equation tells us how much of the drug remains after \( t \) hours. The decay in this case is due to the reduction that occurs with each passing hour, which we can predict using this model.
Using such a formula, we can calculate the concentration at any given time \( t \), and determine when it becomes ineffective.

To model this decay accurately, we follow a step-by-step process to ensure precision in our calculations. Organizing in steps allows us not just to see the progression but also to easily check our work.To start, we rejected the initial concentration and decay rate values into the exponential decay formula, \( C(t) = 640(0.8)^t \). This gives us a clear path to mapping the drug's concentration over time.
Next, creating a table of values helps illustrate the pattern of decay. By calculating \( C(t) \) for each hour, we see how the concentration decreases. We measured values like:

• \( C(0) = 640 \)
• \( C(1) = 512 \)
• \( C(2) = 409.6 \)
• Continue this until you observe the concentration falling below 100 mg/L.

As calculated, between \( t = 8 \) and \( t = 9 \), the concentration slips below the effective threshold, dropping to approximately 85.9 mg/L by \( t = 9 \). Ensuring calculations are listed step-by-step consolidates our understanding and verifies the answer. This methodical approach boosts clarity and helps in replicating results as needed, like adjusting parameters for alternate examples or exact dosing needs.