To determine whether the function represents a direct variation, we need to check two key features:
1. The function must have a constant rate of change.
2. The function must pass through the origin (0, 0).
Let's first calculate the rate of change, also known as the slope. The rate of change can be calculated as the change in cost divided by the change in time.
Using the data points from the table:
- From time [tex]\(0\)[/tex] to [tex]\(2\)[/tex] hours, the cost changes from \[tex]$0 to \$[/tex]10.
- From time [tex]\(2\)[/tex] to [tex]\(4\)[/tex] hours, the cost changes from \[tex]$10 to \$[/tex]20.
- From time [tex]\(4\)[/tex] to [tex]\(6\)[/tex] hours, the cost changes from \[tex]$20 to \$[/tex]30.
- From time [tex]\(6\)[/tex] to [tex]\(8\)[/tex] hours, the cost changes from \[tex]$30 to \$[/tex]40.
The rate of change (slope) for each interval is:
[tex]\[
\frac{\Delta \text{Cost}}{\Delta \text{Time}} = \frac{10 - 0}{2 - 0} = 5 \quad \text{dollars per hour}
\][/tex]
[tex]\[
\frac{\Delta \text{Cost}}{\Delta \text{Time}} = \frac{20 - 10}{4 - 2} = 5 \quad \text{dollars per hour}
\][/tex]
[tex]\[
\frac{\Delta \text{Cost}}{\Delta \text{Time}} = \frac{30 - 20}{6 - 4} = 5 \quad \text{dollars per hour}
\][/tex]
[tex]\[
\frac{\Delta \text{Cost}}{\Delta \text{Time}} = \frac{40 - 30}{8 - 6} = 5 \quad \text{dollars per hour}
\][/tex]
As we can see, the rate of change is constant at [tex]\(\$5\)[/tex] per hour across all intervals.
Next, we need to check if the function passes through the origin. From the table, we see that at time [tex]\(0\)[/tex] hours, the cost is \[tex]$0. Therefore, the function passes through the origin \((0, 0)\).
Since the function has a constant rate of change of \(\$[/tex]5\) per hour and passes through the origin, it indeed represents a direct variation.
Thus, the correct explanation is:
"This function represents a direct variation because it has a positive, constant rate of change of [tex]\( \$5 \)[/tex] per hour."
