One yardstick for measuring how steadily—if slowly

One yardstick for measuring how steadily—if slowly—athletic performance has improved is the mile run. In 1954, Roger Bannister of Britain cracked the 4-minute mark, setting the record for running a mile in 3 minutes, 59.4 seconds, or 239.4 seconds. In the half-century since then, the record has decreased by 0.3 second per year. a. Express the record time for the mile run, M, as a function of the number of years after 1954, x. b. If this trend continues, in which year will someone run a 3-minute, or 180-second, mile?

One yardstick for measuring how steadily—if slowly—athletic performance has improved is the mile run. In 1954, Roger Bannister of Britain cracked the 4-minute mark, setting the record for running a mile in 3 minutes, 59.4 seconds, or 239.4 seconds. In the half-century since then, the record has decreased by 0.3 second per year.
a. Express the record time for the mile run, M, as a function of the number of years after 1954, x.
b. If this trend continues, in which year will someone run a 3-minute, or 180-second, mile?

Key Concepts

Function Modeling

Function modeling involves creating a mathematical expression that accurately represents the relationship between variables based on given data or trends. In this problem, the relationship between the year and the mile run record is modeled using a linear function, which helps to predict future trends based on past performance.

Extrapolation

Extrapolation is the process of using a mathematical model to estimate values beyond the range of the observed data. In this scenario, once the linear function is established, extrapolation allows us to predict when the mile record might reach a specified time (such as 180 seconds) by solving the function for the corresponding year.

Linear Functions

A linear function represents a constant rate of change between two variables, typically expressed in the form y = mx + b, where m is the slope and b is the y-intercept. In this context, the linear function models the decrease in record time over the years, showing how consistently the time decreases each year.

Slope and Intercept

The slope (m) of a linear function indicates the rate at which the dependent variable changes per unit change in the independent variable, while the intercept (b) represents the starting value when the independent variable is zero. Here, the intercept corresponds to the record time in 1954, and the slope is the annual decrease in the record time.

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