A linear function changes by the same amount each minute, so two time-temperature readings determine its rate. Divide the temperature change by the matching time change, then use either reading to find the temperature at time zero. Desmos can calculate the rate and solve for that starting value. The first reported temperature is not necessarily the starting temperature.
Hints
- Hint 1
Each reading pairs a time with a temperature. The slope is the temperature change divided by the time change. Keep the readings in the same order in both parts of that fraction.
- Hint 2
A temperature drop across several minutes is not the drop per minute. Divide by the elapsed time, and keep the sign: a cooling liquid has a negative slope.
- Hint 3
In , the intercept is the temperature when . The first reading happens later than that, so substitute its time and temperature to find the starting value.
Step-by-step
Find the rate, then the starting temperature
Step 1Pair each time with its temperature
The input counts minutes, and the output gives temperature, so the two readings mean . These are the points and .
- Step 2
Find the change per minute
The slope is the change in temperature divided by the change in time. Subtract the readings in the same order: . Type in Desmos; it shows . So the liquid cools by degrees Fahrenheit each minute.
- Step 3
Find the temperature at time zero
A linear function has the form , where is the starting temperature. Use the reading at 5 minutes with the slope you found: . Type in Desmos; under PARAMETERS, it shows . A reported temperature is the intercept only if its time is .
- Step 4
Write the temperature model
Put the negative rate with and the intercept in the constant term: . This gives the modeled temperature in degrees Fahrenheit minutes after removal from the heat source. Choice B.