After a certain medicine is ingested, its concentration in the bloodstream changes over time.

The relationship between the elapsed time, \[t\], in minutes, since the medicine was ingested, and its concentration in the bloodstream, \[C_{\text{minute}}(t)\], in \[\text{mg/L}\], is modeled by the following function:

\[C_{\text{minute}}(t)=61\cdot (0.96)\^{\Large t}\]

**Complete the following sentence about the *hourly* rate of change in the medicine concentration.**
*Round your answer to two decimal places.*

Every ***hour***, the medicine concentration decays by a factor of \[\ \] .

The given function modeling the concentration of medicine in the bloodstream is:

\[
C_{\text{minute}}(t) = 61 \cdot (0.96)^t
\]

Here, \( t \) is the time in minutes, and the base \( 0.96 \) represents the decay factor per minute.

To find the decay factor per hour, we need to convert the minute-based decay factor into an hour-based decay factor. Since there are 60 minutes in an hour, we raise the minute-based decay factor to the power of 60:

\[
\text{Hourly decay factor} = (0.96)^{60}
\]

Let's calculate this value:

\[
\text{Hourly decay factor} \approx (0.96)^{60} \approx 0.2825
\]

So, every hour, the medicine concentration decays by a factor of approximately **0.28** when rounded to two decimal places.

Would you like more details or have any questions?

Here are 5 relative questions to expand your understanding:
1. How would the medicine concentration change after 2 hours?
2. What does the decay factor of 0.28 per hour imply about the medicine's effectiveness?
3. How would the model change if the medicine concentration increased instead of decreased?
4. How could this decay factor be used to determine the best time to administer the next dose?
5. What would be the concentration after 30 minutes?

**Tip:** Exponential decay models are commonly used in pharmacokinetics to describe how drugs are processed in the body over time. Understanding the decay factor helps in dosing schedules.

Explore how medicine concentration in the bloodstream decays over time using an exponential decay model. Learn about the factors influencing decay rates and how to calculate the hourly decay factor. Suitable for advanced high school students.

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