グローバルビジネスで役立つ数学（８）血中の薬物量と時間（英語版）

Hi, there!

Every year, I teach Business English, online and in-person, taking advantage
of my hands-on experience working overseas as an expatriate.

To my consternation, most of my students are engaged in the pharmaceuti-
cal industry. We are inundated with a lot of medical jargon, so we build voca-
bulary in that field, accordingly and unconsciously. I always start with “What’s
new?” talks, sharing their anecdotes with classmates and adding extra voca-
bulary genres, such as medication, which enriches our conversations.

Today, let me give the calculation of the medication a try.

Jennifer has taken an initial dose of a prescription medication. The relation-
ship between the elapsed time \(t\) (hours), since she took the first dose, and
the amount of medication, \(M (t)\) (milligrams), in her bloodstream is model-
ed by the following function:

\(M (t) =10・e^{-0.5t}\). In how many hours will Jennifer have 2 mg of medica-
tion remaining in her bloodstream? Round your answer, if necessary, to the
nearest hundredth.

I do not take any medicine even if I have a cold because I do not want to swa-
llow any foreign object into my body. The word problem above does not refle-
ct my authentic lifestyle, but I have no choice but to give it a shot to beef up
my mathematical skills.

Here is my solution to the problem.

At any rate, I have to know how many hours it will take \((t)\) for the amount of
medication in Jennifer’s bloodstream, \(M (t)\), to drop to 2 mg. So I must find
the value of \(t\) for which \(M (t) = 2\).

Substituting 2 for \(M (t)\) in the function gives me the equation below

\(2 = 10・e^{-0.5t}\).

Then, I can solve the equation as:

\(10・e^{-0.5t} = 2\)
\(e^{-0.5t} = 0.2\)
\(-0.5t = ln (0.2)\)
\(t = \frac{ln(0.2)}{-0.5}\)
\(ln(0.2) = \frac{-1.6094379124341}{-0.5}\)
\(t ≈ 3.22\)

Jennifer will have 2 mg of the medication remaining in his blood after 3.22
hours.

For your reference,

\(ln(x)\) = natural logarithm
\(log(x)\) = common logarithm
\(e\) = Napier’s constant = 2.71828182845

For Japanese mathematics learners, you can use syllabary to memorize
the Napier’s constant like:

“Fu-na-bito-ya-tsu-wa-i-ppa-tsu-ha-shi-go”（2.71828182845）

Let’s chant it like a sutra.

Stay tuned, and expect to see my next post.

Keep well.

Frank Yoshida

￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣
【グローバルビジネスで役立つ数学】でもっと学習する b＾＾)
【参考図書】『もう一度高校数学』（著者：高橋一雄氏）株式会社日本実業出版社
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