Let's solve the problem step by step.
### Step 1: Understand the Data
We are given the following data points that represent the cost of renting a bicycle as a function of time:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Time (hours)} & \text{Cost (\$)} \\
\hline
0 & 0 \\
\hline
2 & 10 \\
\hline
4 & 20 \\
\hline
6 & 30 \\
\hline
8 & 40 \\
\hline
\end{array}
\][/tex]
### Step 2: Check if it Passes Through the Origin
For a function to represent a direct variation, it should pass through the origin (0, 0). From the table, we see that when the time is 0 hours, the cost is also 0 dollars. Thus, the function does pass through the origin.
### Step 3: Calculate the Rate of Change
To determine if the function has a constant rate of change, we will calculate the rate of change between each pair of points and see if it is constant.
The formula for the rate of change (slope) between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Let's apply this to each interval:
- Between (0, 0) and (2, 10):
[tex]\[
\text{Rate of Change} = \frac{10 - 0}{2 - 0} = \frac{10}{2} = 5 \text{ dollars/hour}
\][/tex]
- Between (2, 10) and (4, 20):
[tex]\[
\text{Rate of Change} = \frac{20 - 10}{4 - 2} = \frac{20 - 10}{2} = \frac{10}{2} = 5 \text{ dollars/hour}
\][/tex]
- Between (4, 20) and (6, 30):
[tex]\[
\text{Rate of Change} = \frac{30 - 20}{6 - 4} = \frac{30 - 20}{2} = \frac{10}{2} = 5 \text{ dollars/hour}
\][/tex]
- Between (6, 30) and (8, 40):
[tex]\[
\text{Rate of Change} = \frac{40 - 30}{8 - 6} = \frac{40 - 30}{2} = \frac{10}{2} = 5 \text{ dollars/hour}
\][/tex]
We observe that the rate of change is constant and equals [tex]$5$[/tex] dollars per hour for all intervals.
### Step 4: Conclude the Type of Variation
Since the function passes through the origin and has a constant rate of change (\[tex]$5 per hour), it represents a direct variation. This can be described by the equation \( y = kx \), where \( k \) is the constant of proportionality or the rate of change.
### Final Answer
The correct explanation from the options given 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."
