Question 206·Medium·Nonlinear Functions
A certain medication in the bloodstream decreases in quantity according to the model
where is the amount, in milligrams, remaining hours after the medication is first administered.
According to the model, approximately what percent of the medication is eliminated from the bloodstream each hour?
When you see an exponential model like , interpret as the growth/decay factor for every time units. First, identify the percent remaining (and eliminated) over that -unit interval, then translate it to a per‑unit rate by taking the appropriate root (here, the fourth root for 4 hours). Avoid just dividing the multi‑hour percent change by the number of hours; exponential decay compounds, so you must work with powers and roots, then convert the per‑unit multiplier into a percent removed or remaining, depending on what the question asks.
Hints
Focus on the exponent
Look at the exponent in . What time interval does this suggest the factor applies to?
Check what happens after 4 hours
Substitute into to see what fraction of the medication remains after 4 hours. How much is eliminated in that 4‑hour period?
Go from 4-hour rate to 1-hour rate
If a fraction of the medication remains after 1 hour, then after 4 hours the remaining fraction is . Set equal to the fraction you found for 4 hours and solve for .
Turn the remaining fraction into a percentage eliminated
Once you have the fraction that remains each hour, how do you find the fraction (or percent) that is eliminated each hour?
Desmos Guide
Enter the function
Type f(t) = 120*(0.85)^(t/4) into Desmos so you have the same model as in the problem.
Compute the hourly multiplier
In a new expression line, type f(1)/f(0) or equivalently (0.85)^(1/4); this value is the fraction of medication that remains after each hour.
Find the hourly percent eliminated
In another line, type 1 - (f(1)/f(0)) or 1 - (0.85)^(1/4); read the decimal result and convert it to a percentage, then select the answer choice closest to that percent.
Step-by-step Explanation
Interpret the exponential model
The model is
Here, is the initial amount, and the factor tells how the amount changes over time. The exponent means “number of 4‑hour intervals in hours,” so every 4 hours the amount is multiplied by .
Find the change over 4 hours
If we plug in (4 hours), we get
So after 4 hours, 85% of the medicine remains. That means 15% is eliminated every 4 hours, but the question asks for the percent eliminated each hour.
Relate 4-hour change to 1-hour change
Let be the fraction of medication that remains after 1 hour. After 4 hours, the remaining fraction would be .
From the model, after 4 hours the remaining fraction is , so
We need to approximate . Try :
So is about , meaning about 96% of the medication remains each hour.
Convert remaining fraction to percent eliminated
If about 96% remains each hour, then the percent eliminated each hour is the complement of 96%:
So, approximately 4% of the medication is eliminated from the bloodstream each hour.