To analyze the linear equation \( f(x) = 10.2217 - 0.0098x \), where \( f(x) \) represents the finishing time of the Olympic gold medalist in the men's 100 m dash and \( x \) represents the number of years since 1960, letโs address the parts one by one:
Part A: Interpret the \( y \)-intercept of the linear equation modeling the data.
The \( y \)-intercept occurs when \( x = 0 \). This corresponds to the year 1960 (since \( x \) is the number of years since 1960).
\[ f(0) = 10.2217 - 0.0098(0) = 10.2217 \]
Interpretation: The model suggests that the finishing time of the Olympic gold medalist in the men's 100 m dash in 1960 was 10.2217 seconds.
Part B: Interpret the slope of the linear equation that models the data.
The slope of the line is \(-0.0098\). This value represents the rate of change of the finishing time concerning the years since 1960.
Interpretation: The finishing time since 1960 has decreased each year by 0.0098 seconds.
Part C: The Summer Olympic Games, where the 100 m dash is run, occur every 4 years. By how many seconds does the model predict the first-place finishing time will decrease every 4 years?
To find the total decrease over 4 years, we can multiply the slope by 4:
\[ \text{Decrease over 4 years} = -0.0098 \times 4 = -0.0392 \]
Answer: The model predicts the first-place finishing time will decrease by 0.0392 seconds every 4 years.