Question 6·Hard·Two-Variable Data: Models and Scatterplots
The scatterplot above shows the men’s world-record marathon time, in minutes, for several years between 1980 and 2015. A line of best fit for the data is shown with equation , where is the number of years since 1980 and is the record time in minutes.
According to this model, in which year is the men’s world-record marathon time predicted to first drop below 2 hours?
For line-of-best-fit questions that ask when a value reaches or crosses a certain level, ignore the scatterplot points and focus on the equation given. First, convert units if needed (for example, hours to minutes), then set the model equal to the target value and solve for the variable (here, years since a starting date). Finally, interpret that variable in context—convert to an actual year and use the idea that crossing the target (equal to) happens just before going above or below it, depending on the question. This approach is quick, avoids graph-reading errors, and keeps the algebra simple.
Hints
Match the units
The equation gives the record time in minutes, but the question talks about 2 hours. How many minutes is 2 hours?
Use the line-of-best-fit equation
Once you know the target time in minutes, substitute that value for in the equation and solve for . What does that equation look like?
Interpret the solution
Your solution for will tell you how many years after 1980 the model hits 2 hours. How do you turn that into an actual calendar year?
Connect "equal to" and "below"
If the model reaches exactly 2 hours at some time during a particular year, what can you say about when it first goes below 2 hours, in terms of that same year?
Desmos Guide
Graph the model and the 2-hour line
In Desmos, enter the model as y = -0.27x + 130 and then enter a second line y = 120 to represent 2 hours (120 minutes).
Find the intersection point
Use Desmos’s point-of-intersection feature (or click where the two lines cross) and note the x-coordinate of that point; this is the number of years after 1980 when the model predicts exactly 2 hours.
Convert x to a calendar year
Take the x-value you found and add 1980 to it to get the calendar year. That year is when the model predicts the time will reach 2 hours, and slightly later in that same year the time will be below 2 hours.
Step-by-step Explanation
Convert 2 hours to minutes
The model’s equation uses minutes for , so convert 2 hours to minutes:
minutes.
So we want to know when the predicted record time first goes below 120 minutes.
Set up the equation with the model
The line of best fit is
We want to find when the time reaches 120 minutes, so set :
This gives the point where the model predicts exactly 2 hours; just after this point, the time will be below 2 hours.
Solve for the number of years since 1980
Solve the equation step by step:
Compute the fraction:
So the model predicts the record reaches 120 minutes about 37 years after 1980.
Convert years since 1980 to a calendar year and interpret "below"
If is the number of years since 1980, then
That means the record time hits about 120 minutes sometime during 2017, and then becomes less than 120 minutes later that same year. So according to the model, the record first drops below 2 hours in 2017.