Let's break down the interpretations based on the given linear equation \( f(x) = 10.2217 - 0.0098x \):
Part A: Interpret the \( y \)-intercept of the linear equation.
The \( y \)-intercept occurs when \( x = 0 \). In this context, \( x = 0 \) corresponds to the year 1960. To find the \( y \)-intercept, we substitute \( x = 0 \) into the equation:
\[ f(0) = 10.2217 - 0.0098(0) = 10.2217 \]
This means 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, \( -0.0098 \), indicates the rate of change of the finishing time per year. Specifically, it suggests that for each additional year since 1960, the finishing time decreases 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?
Since the slope indicates the time decreases by \( 0.0098 \) seconds each year, over a span of 4 years, the total decrease can be calculated as:
\[ \text{Decrease in 4 years} = 0.0098 \text{ seconds/year} \times 4 \text{ years} = 0.0392 \text{ seconds} \]
Thus, the model predicts that the first-place finishing time will decrease by 0.0392 seconds every 4 years.