The table represents a bicycle rental cost in dollars as a function of time in hours.
Bicycle Rental
| Time (hours) | Cost ($) |
|---|---|
| 0 | 0 |
| 2 | 10 |
| 4 | 20 |
| 6 | 30 |
| 8 | 40 |
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 (2)
Calculate the rate of change between consecutive data points and observe that it is constant.
Verify that the function passes through the origin (0, 0).
Conclude that the function represents a direct variation because it has a constant rate of change of $5 per hour and passes through the origin.
The correct explanation is: This function represents a direct variation because it passes through the origin and has a constant rate of change of \boxed{$5 per hour}.
Explanation
Understanding Direct Variation We are given a table that shows the cost of renting a bicycle for different amounts of time. We need to determine if this relationship represents a direct variation and choose the correct explanation. A direct variation is a linear relationship where the ratio between two variables is constant, and the graph passes through the origin (0, 0).
Calculating Rate of Change To check if the relationship is a direct variation, we need to calculate the rate of change between consecutive data points. The rate of change is calculated as the change in cost divided by the change in time.
Determining Constant Rate of Change Let's calculate the rate of change between the given data points:
Between (0, 0) and (2, 10): 2 − 0 10 − 0 = 2 10 = 5
Between (2, 10) and (4, 20): 4 − 2 20 − 10 = 2 10 = 5
Between (4, 20) and (6, 30): 6 − 4 30 − 20 = 2 10 = 5
Between (6, 30) and (8, 40): 8 − 6 40 − 30 = 2 10 = 5
The rate of change is constant and equal to $5 per hour.
Conclusion Since the rate of change is constant ($5 per hour) and the function passes through the origin (0, 0), the function represents a direct variation. Therefore, the correct explanation is: This function represents a direct variation because it passes through the origin and has a constant rate of change of $5 per hour.
Examples
Direct variation is a fundamental concept in many real-world scenarios. For instance, the relationship between the number of hours you work and the amount you earn, if you are paid a fixed hourly wage, is a direct variation. If you earn $15 per hour, the more hours you work, the more money you earn, and the relationship is linear and passes through the origin (0 hours, $0 earned). Similarly, the distance you travel at a constant speed is directly proportional to the time you travel. These examples illustrate how direct variation helps us understand and predict relationships in everyday situations.
The bicycle rental cost function represents direct variation because it passes through the origin and maintains a constant rate of change of $5 per hour. Therefore, the correct answer is option A. This confirms the relationship between time and cost follows a linear pattern.
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