Question 30·Hard·Two-Variable Data: Models and Scatterplots
A scientist studying the breakdown of a medication recorded the amount of the medication, (in milligrams), remaining in a patient’s bloodstream at various times, (in hours). She plotted versus and found the line of best fit
Based on this model, what is the predicted amount of the medication, in milligrams, remaining after 12 hours? Round your answer to the nearest whole number.
When a question gives a linear equation for (or any logarithm of a quantity) and asks for the original quantity, first substitute the given time or input into the linear model to find the log value. Then, remember that logs and exponentials are inverse operations: convert back by exponentiating with the correct base (here, ). Use your calculator’s or exp function to evaluate, and only round to the nearest whole number at the very end to avoid errors.
Hints
Substitute the given time
You are given a formula for in terms of . First, plug in to find after 12 hours.
Work carefully with the arithmetic
Compute and then subtract that result from to get a numerical value for .
Undo the natural log
Once you have (some number), remember that is a logarithm with base . What operation reverses taking a natural log so you can get itself?
Use your calculator wisely
After you rewrite in terms of an exponential expression, use the or exp button to evaluate it, then round to the nearest whole number.
Desmos Guide
Evaluate the exponential expression
In Desmos, type exp(5.30 - 0.18*12) or e^(5.30 - 0.18*12) as a single expression. Read off the numeric output that Desmos gives you, then round that value to the nearest whole number to match one of the answer choices.
Step-by-step Explanation
Use the line of best fit to find at 12 hours
The model is
We want the amount of medication after 12 hours, so substitute :
Now calculate .
Compute the numerical value of
Evaluate the product:
So
This means the natural log of after 12 hours is .
Undo the natural logarithm to solve for
The equation means that is the number whose natural log is . To undo a natural log, exponentiate both sides with base :
Now we just need to approximate .
Approximate and round
Use a calculator for (or the exp(3.14) function):
Rounding to the nearest whole number gives milligrams.
So, the predicted amount of medication remaining after 12 hours is 23 milligrams.