To determine whether the function represents a direct variation, we need to check two conditions:
1. The function must pass through the origin (0,0).
2. The function must have a constant rate of change (slope).
Let's examine these conditions step by step:
### Step 1: Check if the Function Passes Through the Origin
From the table:
- At [tex]\( t = 0 \)[/tex] hours, the cost is \[tex]$0.
Since the point (0,0) is included in the table, the function passes through the origin.
### Step 2: Check the Rate of Change
Next, we need to determine the rate of change and see if it is consistent:
- From \( t = 0 \) hours to \( t = 2 \) hours, the cost changes from \$[/tex]0 to \[tex]$10.
\[
\text{Rate of change} = \frac{\$[/tex]10 - \[tex]$0}{2 - 0 \text{ hours}} = \frac{10}{2} = 5 \text{ dollars per hour}
\]
- From \( t = 2 \) hours to \( t = 4 \) hours, the cost changes from \$[/tex]10 to \[tex]$20.
\[
\text{Rate of change} = \frac{\$[/tex]20 - \[tex]$10}{4 - 2 \text{ hours}} = \frac{10}{2} = 5 \text{ dollars per hour}
\]
- From \( t = 4 \) hours to \( t = 6 \) hours, the cost changes from \$[/tex]20 to \[tex]$30.
\[
\text{Rate of change} = \frac{\$[/tex]30 - \[tex]$20}{6 - 4 \text{ hours}} = \frac{10}{2} = 5 \text{ dollars per hour}
\]
- From \( t = 6 \) hours to \( t = 8 \) hours, the cost changes from \$[/tex]30 to \[tex]$40.
\[
\text{Rate of change} = \frac{\$[/tex]40 - \[tex]$30}{8 - 6 \text{ hours}} = \frac{10}{2} = 5 \text{ dollars per hour}
\]
The rate of change is constant at 5 dollars per hour.
### Conclusion
Since the function passes through the origin and has a constant rate of change of 5 dollars per hour, we can conclude that the function represents a direct variation.
The correct explanation is:
This function represents a direct variation because it passes through the origin and has a constant rate of change of \$[/tex]5 per hour.
