To determine the equation that represents the variation of cost with the number of minutes Jalen talks on the phone, we need to verify if the cost varies directly with the time. If the cost varies directly with the time spent on calls, then [tex]\( y \)[/tex] (cost) is proportional to [tex]\( x \)[/tex] (minutes), and the equation would be in the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant rate of change (slope).
To find this constant rate of change [tex]\( k \)[/tex]:
1. Calculate the rate of change (slope) between pairs of data points:
The pairs of data points are:
- (150, \[tex]$7.50) and (220, \$[/tex]11.00)
- (220, \[tex]$11.00) and (250, \$[/tex]12.50)
- (250, \[tex]$12.50) and (275, \$[/tex]13.75)
The slope between each pair of points is given by:
[tex]\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
2. Compute the slope for each pair:
- Between (150, \[tex]$7.50) and (220, \$[/tex]11.00):
[tex]\[
\text{slope}_1 = \frac{11.00 - 7.50}{220 - 150} = \frac{3.50}{70} = 0.05
\][/tex]
- Between (220, \[tex]$11.00) and (250, \$[/tex]12.50):
[tex]\[
\text{slope}_2 = \frac{12.50 - 11.00}{250 - 220} = \frac{1.50}{30} = 0.05
\][/tex]
- Between (250, \[tex]$12.50) and (275, \$[/tex]13.75):
[tex]\[
\text{slope}_3 = \frac{13.75 - 12.50}{275 - 250} = \frac{1.25}{25} = 0.05
\][/tex]
3. Verify that the slopes are equal:
Since [tex]\( \text{slope}_1 = 0.05 \)[/tex], [tex]\( \text{slope}_2 = 0.05 \)[/tex], and [tex]\( \text{slope}_3 = 0.05 \)[/tex], we can conclude that the cost indeed varies directly with the number of minutes because the rate of change is consistent across all pairs of points.
4. Formulate the equation:
Given that the rate of change [tex]\( k = 0.05 \)[/tex], the equation representing the direct variation is:
[tex]\[
y = 0.05x
\][/tex]
Therefore, the correct equation that represents the variation of cost with the number of minutes Jalen talks is:
[tex]\[
\boxed{y = 0.05x}
\][/tex]
