Let's determine whether the given data represents a direct variation by examining its properties.
First, recall that a function represents a direct variation if:
1. The graph passes through the origin (0, 0).
2. The function has a constant rate of change (slope).
Here are the given data points:
- Time (hours): 0, 2, 4, 6, 8
- Cost (dollars): 0, 10, 20, 30, 40
### Step-by-Step Analysis
1. Passes through the origin:
- We check the point (0, 0) in the table:
- Time = 0 hours, Cost = \[tex]$0
- Since the function passes through the origin (0, 0), this condition is satisfied.
2. Constant rate of change:
- Calculate the rate of change between consecutive data points:
\[
\text{Rate of Change} = \frac{\text{Change in Cost}}{\text{Change in Time}}
\]
- Between (0, 0) and (2, 10):
\[
\frac{10 - 0}{2 - 0} = \frac{10}{2} = 5
\]
- Between (2, 10) and (4, 20):
\[
\frac{20 - 10}{4 - 2} = \frac{10}{2} = 5
\]
- Between (4, 20) and (6, 30):
\[
\frac{30 - 20}{6 - 4} = \frac{10}{2} = 5
\]
- Between (6, 30) and (8, 40):
\[
\frac{40 - 30}{8 - 6} = \frac{10}{2} = 5
\]
- The rate of change is constant and equals \$[/tex]5 per hour.
Since the function passes through the origin and has a constant rate of change of \[tex]$5 per hour, it satisfies the conditions of a direct variation.
### Conclusion
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.
