Question 372·Hard·Algebra

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The concentration $C$ , in milligrams per liter, of a medication in a patient’s bloodstream is modeled as a linear function of time $t$ , in hours, after the medication is administered. Three hours after administration, the concentration is 52 mg/L, and eleven hours after administration, the concentration is 28 mg/L.

According to this model, after how many hours will the concentration reach 20 mg/L?

A linear model has a constant change per hour, so two time-concentration pairs determine its rate and starting value. Enter the pairs in a Desmos table and fit a line, then use a regression to find the time for the requested concentration. Keep the quantities straight: concentration is the output, not the number of hours.

Hints

Hint 1
A linear function changes by the same amount each hour. Pair each time with its concentration to make two points. Which number in each pair is the input?
Hint 2
The slope is the change in concentration divided by the change in time. Since concentration falls as time passes, the slope must be negative.
Hint 3
The target concentration is an output, not an hour. Once you have the line, set its output equal to the target and solve for the time that produces it.

Step-by-step

Fit the line, then find the time
Step 1
Pair each time with its concentration
Time $t$ is the input, and concentration $C$ is the output, so the measurements give the points $(t,C)=(3,52)$ and $(11,28)$ . Keep each time with its matching concentration; these are two points on the same line.
Step 2
Find the line's hourly change
Enter the points in a Desmos table, with time in $x_1$ and concentration in $y_1$ . Then type $y_1\sim mx_1+b$ ; a regression fits the line through the points. Under PARAMETERS, Desmos shows $m=-3$ and $b=61$ . The negative slope means the concentration drops $3$ mg/L each hour.
Step 3
Write the concentration model
In $f(t)=mt+b$ , the slope $m$ is the hourly change and $b$ is the value at hour $0$ . Using the values Desmos found gives $f(t)=-3t+61$ , where $f(t)$ is the modeled concentration. The starting value is $61$ , not $52$ : the $52$ mg/L measurement was taken at hour $3$ .
Step 4
Solve for the time at 20 mg/L
Set the output equal to the requested concentration: $f(t)=20$ . Type $f(t)=-3t+61$ , then $f(t_1)\sim20$ ; the subscript makes $t_1$ the time Desmos solves for. Under PARAMETERS, it shows $t_1\approx13.6667$ . Type $t_1$ on the next line and use the fraction button to see $\frac{41}{3}$ . So the concentration reaches $20$ mg/L after $\frac{41}{3}$ hours. Choice B.

Question 372·Hard·Algebra

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Step-by-step

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