Question 687·Hard·Algebra

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During a laboratory test, the temperature $T$ , in degrees Celsius, of a heated solution is predicted by the linear model

T(t)=90-2\bigl(t-3\bigr),

where $t$ is the time, in minutes, after the test begins. For safety reasons, the experiment must be stopped when the temperature first reaches exactly $60$ degrees Celsius.

According to the model, how many minutes after the test begins should the experiment be stopped?

A linear model changes by the same amount each minute. If a question gives the temperature you’re waiting for, set the model equal to that temperature and solve for time. Desmos can solve that equation directly. First check what time the model’s stated temperature belongs to: a shift inside parentheses may mean it isn’t the temperature at the start.

Hints

Hint 1
The expression $t-3$ counts minutes since minute $3$ . In a linear model, the number outside the parentheses changes the temperature for each of those minutes. What temperature does the model give when $t=3$ ?
Hint 2
The stopping condition gives an output, not an input: $T(t)$ must equal $60$ . Write that equation, then solve for $t$ , which measures minutes after the test begins.

Step-by-step

Set the target temperature and solve in Desmos
Step 1
Identify the model’s starting point
In the given linear model, $t-3$ counts minutes since minute $3$ . At $t=3$ , that part is $0$ , so the model predicts $90$ degrees Celsius then. The $90$ -degree reading belongs to minute $3$ , not minute $0$ .
Step 2
Turn the stopping condition into an equation
The experiment stops when $T(t)$ , the predicted temperature at minute $t$ , is exactly $60$ . Set the model equal to that temperature:
$90-2(t-3)=60.$
The rate of change is $-2$ degrees per minute, so the model is cooling and reaches $60$ only once.
Step 3
Find the minute when the model reaches 60
Type $T(t)=90-2(t-3)$ , then $T(t_{1})\sim60$ . The subscript makes $t_{1}$ a value for Desmos to find, and $\sim$ tells it to match the target temperature. Under PARAMETERS, Desmos shows $t_{1}=18$ . Since $t$ measures minutes after the test begins, stop the experiment after $18$ minutes. Choice C.

Question 687·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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