To interpret the linear model provided, we can break down the components as follows:
Part A: Interpret the y-intercept of the linear equation.
The y-intercept of a linear equation in the form \( f(x) = mx + b \) is the value of \( f(x) \) when \( x = 0 \). In the equation \( f(x) = 10.2217 - 0.0098x \), when \( x = 0 \) (which corresponds to the year 1960), the finishing time \( f(0) = 10.2217 \) seconds.
Answer: 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 linear equation, represented by \( -0.0098 \), signifies the change in the finishing time for each year since 1960. In this context, it indicates that the finishing time decreases by \( 0.0098 \) seconds each year.
Answer: The finishing time since 1960 has decreased each year by 0.0098 seconds.
Part C: The Summer Olympic Games occur every 4 years.
To determine how much the model predicts the finishing time will decrease every 4 years, we multiply the yearly decrease (the slope) by the number of years in 4 years.
\[ \text{Decrease over 4 years} = 4 \times (-0.0098) = -0.0392 \text{ seconds} \]
Answer: The model predicts the first-place-finishing time will decrease every 4 years by 0.0392 seconds.