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The table represents a bicycle rental cost in dollars as a function of time in hours:

Bicycle Rental
\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Time \\
(hours)
\end{tabular} & \begin{tabular}{c}
Cost \\
(\[tex]$)
\end{tabular} \\
\hline
0 & 0 \\
\hline
2 & 10 \\
\hline
4 & 20 \\
\hline
6 & 30 \\
\hline
8 & 40 \\
\hline
\end{tabular}

Which explains whether or not the function represents a direct variation?

A. This function represents a direct variation because it passes through the origin and has a constant rate of change of $[/tex]\[tex]$ 5$[/tex] per hour.
B. This function represents a direct variation because it has a positive, constant rate of change of [tex]$\$[/tex] 10[tex]$ per hour.
C. This function does not represent a direct variation because it does not represent the cost for 1 hour.
D. This function does not represent a direct variation because the function rule for the cost is to add $[/tex]\[tex]$ 10$[/tex], not multiply by a constant.



Answer :

GinnyAnswer
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."

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