A line of best fit predicts a trend, not the exact value of every plotted dot. To find when its prediction first falls below a limit, convert the limit to the model’s units and use a Desmos regression to find where the line reaches it. Then convert elapsed years to a calendar year. The crossing can happen partway through a year.
Hints
- Hint 1
Match units before comparing times. An hour contains minutes, so multiply the hour limit by . The result can then be compared with , which the model measures in minutes.
- Hint 2
The threshold is where the line exactly meets the time limit. Find that crossing first. Since this line slopes downward, its predictions are below the limit after the crossing.
- Hint 3
The model’s is elapsed time, not a calendar year. Add the starting year to the crossing time. If you get a decimal year, which calendar year contains it?
Step-by-step
Find the model’s crossing time
Step 1Convert the limit to minutes
The model measures record times in minutes, so convert the 2-hour limit to the same unit. An hour contains minutes, so minutes.
- Step 2
Write the below-limit condition
Below means strictly less than, so the model’s predicted time must satisfy . The negative coefficient means the prediction falls as the years pass.
- Step 3
Find where the line reaches the limit
First find the boundary, where the prediction equals . Type in Desmos. Use because Desmos fits its value, while it graphs ; asks Desmos to make the two sides equal. Under PARAMETERS, it shows . The prediction reaches minutes a little more than years after 1980, and falls below it after that.
- Step 4
Convert the crossing time to a year
Type on the next Desmos line. It prints about , so the line reaches 2 hours partway through 2017. At the crossing the prediction equals the limit; immediately afterward, it is below. According to the model, the time first drops below 2 hours during 2017. Choice B.