There are 60 minutes in one hour. The total number of minutes is a function of the number of hours, as shown in the table. Does this situation represent a linear or non-linear function, and why?
| Hours | Minutes |
|---|---|
| 1 | 60 |
| 2 | 120 |
| 3 | 180 |
| 4 | 240 |
| 5 | 300 |
A. It represents a linear function because there is a constant rate of change.
B. It represents a linear function because there is not a constant rate of change.
C. It represents a non-linear function because there is a constant rate of change.
D. It represents a non-linear function because there is not a constant rate of change.
Answer (1)
Calculate the rate of change between consecutive data points.
Observe that the rate of change is constant and equals to 60.
Conclude that the relationship represents a linear function because there is a constant rate of change.
The correct answer is A.
Explanation
Understanding the Problem We are given a table that shows the relationship between the number of hours and the corresponding number of minutes. We need to determine whether this relationship represents a linear or non-linear function and provide the correct reason.
Checking for Constant Rate of Change To determine if the relationship is linear, we need to check if there is a constant rate of change between consecutive data points. The rate of change is calculated as the change in minutes divided by the change in hours.
Calculating the Rate of Change Let's calculate the rate of change between the given data points:
Between (1, 60) and (2, 120): Rate of change = 2 − 1 120 − 60 = 1 60 = 60
Between (2, 120) and (3, 180): Rate of change = 3 − 2 180 − 120 = 1 60 = 60
Between (3, 180) and (4, 240): Rate of change = 4 − 3 240 − 180 = 1 60 = 60
Between (4, 240) and (5, 300): Rate of change = 5 − 4 300 − 240 = 1 60 = 60 Since the rate of change is constant (60 minutes per hour), the relationship represents a linear function.
Conclusion Since the rate of change is constant, the function is linear. Therefore, the correct answer is: It represents a linear function because there is a constant rate of change.
Examples
Understanding linear functions is crucial in everyday life. For instance, if you're saving money at a constant rate, the relationship between the time you save and the total amount saved is linear. Similarly, if you're driving at a constant speed, the relationship between the time you drive and the distance you cover is linear. Recognizing these linear relationships helps in making predictions and informed decisions.
