North Carolina School of Science and Mathematics
Distance Education and Extended Programs
Headache Medicine Problem and Exercises
You have a headache and decide to take a 200-mg over-the-counter headache medicine tablet to
relieve it. The ibuprofen in the pill is absorbed into your system and stays there until it is
metabolized and filtered out by the liver and kidneys. Every 4 hours, your body removes 67% of
the ibuprofen that was in your body at the beginning of that 4-hour time period.
1. Write a recursive system that represents the amount of ibuprofen in your body at any time.
How much of the ibuprofen will have remained in your system 24 hours after taking the
200-mg tablet?
{
𝑫 = 𝟐𝟎𝟎
𝑫𝒏 = 𝑫𝒏−𝟏 − 𝟎. 𝟔𝟕𝑫𝒏−𝟏 ,
𝒏≥𝟏
where 𝐧 represents the number of 4-hour intervals since you took the initial dose.
So, since 𝐃𝟔 = 𝟎. 𝟐𝟓𝟖, you will only have 0.258 mg of ibuprofen left in your body
24 hours after taking your dose.
2. Suppose that the tablet contained 250 mg instead of 200. How much would be in your body
after 24 hours?
{
𝑫 = 𝟐𝟓𝟎
𝑫𝒏 = 𝑫𝒏−𝟏 − 𝟎. 𝟔𝟕𝑫𝒏−𝟏 ,
𝒏≥𝟏
where 𝒏 represents the number of 4-hour intervals since you took the initial dose.
So, since 𝑫𝟔 = 𝟎. 𝟑𝟐𝟑, you will only have 0.323 mg of ibuprofen left in your body
24 hours after taking your dose.
3. Suppose that your kidneys filter only 50% of the ibuprofen in 4 hours. If you take a 200-mg
tablet, how much would you have in your body after 24 hours?
{
𝑫 = 𝟐𝟎𝟎
𝑫𝒏 = 𝑫𝒏−𝟏 − 𝟎. 𝟓𝑫𝒏−𝟏 ,
𝒏≥𝟏
where 𝒏 represents the number of 4-hour intervals since you took the initial dose.
So, since 𝑫𝟔 = 𝟑. 𝟏𝟐𝟓, you will only have 3.125 mg of ibuprofen left in your body
24 hours after taking your dose.
4. Write a sentence to interpret the the following recursive systems in the context of this
problem. Find how much ibuprofen remains after 24 hours. The domain of 𝐷𝑛 is 𝑛 =
1, 2, 3, …
Adapted from Contemporary Precalculus through Applications by the North Carolina School of Science and
Mathematics, Everyday Learning Corporation, 2000
North Carolina School of Science and Mathematics
Distance Education and Extended Programs
a. 𝐷0 = 300,
𝐷𝑛 = 𝐷𝑛−1 − 0.8𝐷𝑛−1
You take a 300 mg tablet, and your kidneys filter 80% of the ibuprofen
every 4 hours.
𝑫𝟔 = 𝟎. 𝟎𝟏𝟗𝟐 𝐦𝐠
b. 𝐷0 = 150,
𝐷𝑛 = 𝐷𝑛−1 − 0.2𝐷𝑛−1
You take a 150 mg tablet, and your kidneys filter 20% of the ibuprofen
every 4 hours.
𝑫𝟔 = 𝟑𝟗. 𝟑𝟐𝟏𝟔 𝐦𝐠
c. 𝐷0 = 500,
𝐷𝑛 = 0.2𝐷𝑛−1
You take a 500 mg tablet, and your kidneys filter 80% of the ibuprofen
every 4 hours.
𝑫𝟔 = 𝟎. 𝟎𝟑𝟐 𝐦𝐠
Adapted from Contemporary Precalculus through Applications by the North Carolina School of Science and
Mathematics, Everyday Learning Corporation, 2000