An antibiotic loses 15% of its effectiveness in each hour after it has been taken. The doctor prescribed a 500mg dose; the first dose was taken at 9am.  Construct a model to represent the amount of antibiotic remaining after any given number of hours, then use your model to determine the amount of antibiotic that remains at 8:00pm.  *Round your answer to the nearest tenth of a milligram.*

To construct a model representing the amount of antibiotic remaining after any given number of hours, we can use the exponential decay formula:

\[
A(t) = A_0 (1 - r)^t
\]

Where:
- \( A(t) \) is the amount of antibiotic remaining after \( t \) hours.
- \( A_0 \) is the initial amount of antibiotic (500 mg in this case).
- \( r \) is the decay rate (15%, or 0.15 per hour).
- \( t \) is the number of hours since the antibiotic was taken.

Thus, the model becomes:

\[
A(t) = 500(1 - 0.15)^t = 500(0.85)^t
\]

Now, we need to find the amount of antibiotic remaining at 8:00 PM. Since the first dose was taken at 9:00 AM, the time elapsed from 9:00 AM to 8:00 PM is 11 hours. Substituting \( t = 11 \) into the model:

\[
A(11) = 500(0.85)^{11}
\]

I'll calculate this value.

After 11 hours, approximately 83.7 mg of the antibiotic remains in the system.

Would you like further details or have any other questions?

Here are some related questions:
1. How does the decay rate affect the amount of antibiotic remaining after 5 hours?
2. What would the remaining amount be if the decay rate were 10% instead of 15%?
3. How would the model change if the initial dose were 1000 mg?
4. Can you derive the time it takes for the antibiotic to reduce to half of its initial dose?
5. How can this model be applied to other medications with different decay rates?

**Tip:** Exponential decay models are commonly used in fields like pharmacology and finance to represent processes that decrease over time at a constant percentage.

An antibiotic loses 15% of its effectiveness in each hour after it has been taken. The doctor prescribed a 500mg dose; the first dose was taken at 9am. Construct a model to represent the amount of antibiotic remaining after any given number of hours, then use your model to determine the amount of antibiotic that remains at 8:00pm. Round your answer to the nearest tenth of a milligram.

Learn how to model the effectiveness of a 500mg antibiotic dose over time using an exponential decay formula. This example explores how a 15% decay rate affects the remaining amount of antibiotic in the body after 11 hours, calculated to be approximately 83.7 mg. Suitable for students in Grades 10-12.

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