Algebra 2 Problems of the Day (January 2020)

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January 2020, Part IV

The Question in Part IV is worth 6 credits. Work must be shown. Partial credit is given.


37. Sarah is fighting a sinus infection. Her doctor prescribed a nasal spray and an antibiotic to fight the infection. The active ingredients, in milligrams, remaining in the bloodstream from the nasal spray, n(t), and the antibiotic, a(t), are modeled in the functions below, where t is the time in hours since the medications were taken.

Determine which drug is made with a greater initial amount of active ingredient. Justify your answer.

Sarah’s doctor told her to take both drugs at the same time. Determine algebraically the number of hours after taking the medications when both medications will have the same amount of active ingredient remaining in her bloodstream.

Answer:
To find the initial amounts of the active ingredient, calculate each function for t = 0.
The nasal spray will have 21/15 milligrams, and the antibiotic will have 3 milligrams, so there will be more active ingredient in the antibiotic. (See image below)

To find the number of hours when the two active ingredients will be at the same level, set the two equations equal to each other.
You can see that the quadratic factors into (t + 5)(t + 3), so the entire equation can be multiplied by both binomials to remove the denominators.
After that, you can solve the resulting quadratic equation, getting t = 8 and t = -3. We discard the negative because negative time makes no sense.
They will have the same amount of active ingredient after 8 hours. (See image below)

Comments and questions welcome.

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