To find the equation that represents the total number of riders over time, we need to use the slope-intercept form of the equation of a line, which is given by:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( y \)[/tex] is the total number of riders,
- [tex]\( x \)[/tex] is the number of hours into Tara's shift,
- [tex]\( m \)[/tex] is the slope of the line, and
- [tex]\( b \)[/tex] is the y-intercept.
1. Determine the Slope (m):
To calculate the slope [tex]\( m \)[/tex], we use two points from the data provided. The slope formula is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the first and last points from the table:
- At [tex]\( x_1 = 0 \)[/tex] hours, [tex]\( y_1 = 132 \)[/tex] riders,
- At [tex]\( x_2 = 3 \)[/tex] hours, [tex]\( y_2 = 231 \)[/tex] riders,
Plug these values into the slope formula:
[tex]\[ m = \frac{231 - 132}{3 - 0} = \frac{99}{3} = 33 \][/tex]
So, the slope [tex]\( m \)[/tex] is 33.
2. Determine the y-intercept (b):
The y-intercept [tex]\( b \)[/tex] is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]. From the table, when [tex]\( x = 0 \)[/tex], [tex]\( y = 132 \)[/tex]. Hence, the y-intercept [tex]\( b \)[/tex] is 132.
3. Formulate the Equation:
Now that we have the slope [tex]\( m = 33 \)[/tex] and the y-intercept [tex]\( b = 132 \)[/tex], we can write the equation in slope-intercept form:
[tex]\[ y = 33x + 132 \][/tex]
Thus, the equation that represents the total number of riders over time is:
[tex]\[ y = 33x + 132 \][/tex]
