Question 629·Hard·Math

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The concentration $C(t)$ , in milligrams per liter, of a dye in a solution decreases linearly with time $t$ , measured in minutes after mixing begins. Measurements show that $C(5)=37\quad\text{and}\quad C(12)=16$ . According to this model, after how many minutes from the start of the experiment will the concentration reach $0$ milligrams per liter?

A linear model changes by the same amount each minute, so two time-and-concentration readings determine its rate and starting level. Enter the readings in a Desmos table and fit a line. Then graph the line with $y=0$ and read the time where they meet. Don't confuse minutes after a measurement with minutes from the start.

Hints

Hint 1
A point $(t,C(t))$ keeps a time paired with its concentration. Put the measured times in one Desmos table column and their matching concentrations beside them so the fitted line uses both readings.
Hint 2
The slope tells you how much the concentration changes each minute. A negative slope means it falls. The line's constant term is its concentration at time $0$ , not at the first measurement.
Hint 3
An intercept is where a graph crosses an axis. To find when the concentration reaches zero, look where its line crosses $y=0$ and read the time coordinate, not the concentration coordinate.

Step-by-step

Fit the line and find where it reaches zero
Step 1
Pair each time with its concentration
A coordinate pair $(t,C(t))$ means (minutes since mixing, milligrams per liter). So the readings give $(5,37)$ and $(12,16)$ . Keep each time with its own concentration.
Step 2
Fit the linear model
Enter those pairs in a Desmos table, with time in $x_1$ and concentration in $y_1$ . Because the concentration decreases linearly, type $y_1\sim mx_1+b$ to fit $C(t)=mt+b$ . Under PARAMETERS, Desmos shows $m=-3$ and $b=52$ : the concentration drops $3$ milligrams per liter each minute, and $C(0)=52$ . The $37$ is the level at minute $5$ , not at the start.
Step 3
Set the concentration to the target
The fitted line is $C(t)=52-3t$ . The question asks for the time when the concentration reaches $0$ , so set its output to $0$ :
$0=52-3t.$
Set the output to the target, then find the input.
Step 4
Read the time at the crossing
Type $y=52-3x$ and $y=0$ , using $x$ for minutes since mixing. Click their intersection, the point where both lines have the same height. Desmos shows about $(17.333,0)$ , so the concentration reaches zero about $17.33$ minutes from the start. Choice B.

Question 629·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

Lessons that teach this