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

Bicycle Rental

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

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 \$5 per hour.

B. This function represents a direct variation because it has a positive, constant rate of change of \$10 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 \$10, not multiply by a constant.



Answer :

Sure! Let's determine whether the function representing the bicycle rental cost as a function of time in hours represents a direct variation. Here's a step-by-step analysis:

1. Definition of Direct Variation: A function represents a direct variation if it can be described by the equation \( y = kx \), where \( k \) is a constant. This means that the function passes through the origin (0,0) and has a constant rate of change (slope).

2. Given Values:
- Time (hours): \(0\), \( 2\), \( 4\), \( 6\), \( 8\)
- Cost (\$): \(0\), \( 10\), \( 20\), \( 30\), \( 40\)

3. Checking if it Passes Through the Origin:
- The function must have a cost of \$0 at time 0 hours.
- From the table, we see that at time 0 hours, the cost is \$0.

4. Calculating the Rate of Change:
- The rate of change is the change in cost divided by the change in time (the slope).
- Calculate the rate of change between consecutive points:
- From 0 to 2 hours: \( \frac{10 - 0}{2 - 0} = \frac{10}{2} = 5 \) dollars per hour
- From 2 to 4 hours: \( \frac{20 - 10}{4 - 2} = \frac{10}{2} = 5 \) dollars per hour
- From 4 to 6 hours: \( \frac{30 - 20}{6 - 4} = \frac{10}{2} = 5 \) dollars per hour
- From 6 to 8 hours: \( \frac{40 - 30}{8 - 6} = \frac{10}{2} = 5 \) dollars per hour
- The rate of change is consistent and equal to 5 dollars per hour.

5. Conclusion:
- Since the function passes through the origin and has a constant rate of change of $5 per hour, it represents a direct variation.

Thus, the correct explanation is:
1. This function represents a direct variation because it passes through the origin and has a constant rate of change of $5 per hour.