Question

the graph shows the winning times (in minutes) of a certain international track race, together with a linear approximation of the data. (a) an equation for a linear model, based on data from 1912–2016 (where x represents the year), is y = -0.01721x + 47.35. determine the slope. what does the slope represent? why is the slope negative? (b) what might be the reason there are no data points for the years 1916, 1940, and 1944? (c) the winning time for the 2000 race was 13.22 minutes. what does the model predict, and how far is the prediction from the actual value? (a) the slope of the line is -0.01721. (type an integer or decimal rounded to five decimal places as needed.) the slope indicates that the winning time for the race by an average of minutes each year from 1912 to 2016. the slope is negative because the winning times generally as time progresses. (type an integer or decimal rounded to five decimal places as needed.)

Explanation:

Step1: Analyze the linear equation form

The linear model is given by \( y = -0.01721x + 47.35 \), which is in the slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. So, by comparing with the general form, we can see that the slope \( m=- 0.01721 \).

Step2: Interpret the slope

In the context of this problem, \( x \) represents the year and \( y \) represents the winning time (in minutes) of the race. The slope \( m=\frac{\Delta y}{\Delta x} \), which means that for a unit increase in \( x \) (i.e., for each one - year increase in the year), the change in \( y \) (the winning time) is equal to the slope. Since the slope is negative (\( - 0.01721 \)), it indicates that the winning time for the race decreases by an average of \( 0.01721 \) minutes each year from 1912 to 2016. The slope is negative because the winning times generally decrease as time (the year) progresses.

Answer:

The slope of the line is \(-0.01721\). The slope indicates that the winning time for the race \(\boldsymbol{\text{decreases}}\) by an average of \(\boldsymbol{0.01721}\) minutes each year from 1912 to 2016. The slope is negative because the winning times generally \(\boldsymbol{\text{decrease}}\) as time progresses.