The data in the table below shows the relationship between the total number of houses built in an area and the number of months that passed.
| Months Passed | Total Houses Built |
|---------------|--------------------|
| 0 | 0 |
| 3 | 33 |
| 4 | 46 |
| 8 | 108 |
Which best describes the data set?
A. It is nonlinear because the "Total Houses Built" column does not increase at a constant additive rate.
B. It is nonlinear because the "Months Passed" column does not increase at a constant additive rate.
C. It is nonlinear because the increase in the "Total Houses Built" compared to the increase in the "Months Passed" does not show a constant rate of change.
D. It is linear because the increase in the "Total Houses Built" compared to the increase in the "Months Passed" shows a constant rate of change.
Answer (2)
The data set is nonlinear because the increase in the 'Total Houses Built' compared to the increase in the 'Months Passed' does not show a constant rate of change. This conclusion is based on the varying rates of change calculated between the data points. Therefore, option C is the correct choice.
;
Calculate the rate of change between consecutive points: (0, 0) and (3, 33), (3, 33) and (4, 46), (4, 46) and (8, 108).
The rate of change between (0, 0) and (3, 33) is 11.
The rate of change between (3, 33) and (4, 46) is 13.
The rate of change between (4, 46) and (8, 108) is 15.5.
Since the rates of change are not constant, the data set is nonlinear. The final answer is: It is nonlinear because the increase in the "Total Houses Built" compared to the increase in the "Months Passed" does not show a constant rate of change.
Explanation
Understanding the Problem We are given a table that shows the relationship between the number of months passed and the total number of houses built in an area. We need to determine whether the data set is linear or nonlinear and provide the correct reason.
Checking for Constant Rate of Change To determine if the data set is linear, we need to check if the rate of change between consecutive points is constant. The rate of change is calculated as the change in the total number of houses built divided by the change in the number of months passed.
Rate of Change between (0, 0) and (3, 33) Let's calculate the rate of change between the first two points (0, 0) and (3, 33): 3 − 0 33 − 0 = 3 33 = 11
Rate of Change between (3, 33) and (4, 46) Now, let's calculate the rate of change between the second and third points (3, 33) and (4, 46): 4 − 3 46 − 33 = 1 13 = 13
Rate of Change between (4, 46) and (8, 108) Next, let's calculate the rate of change between the third and fourth points (4, 46) and (8, 108): 8 − 4 108 − 46 = 4 62 = 15.5
Conclusion The rates of change are 11, 13, and 15.5. Since the rates of change are not constant, the data set is nonlinear. Therefore, the increase in the "Total Houses Built" compared to the increase in the "Months Passed" does not show a constant rate of change.
Final Answer The data set is nonlinear because the increase in the "Total Houses Built" compared to the increase in the "Months Passed" does not show a constant rate of change.
Examples
Understanding whether a relationship is linear or nonlinear is crucial in many real-world scenarios. For instance, when analyzing the growth of a plant over time, the relationship between the amount of fertilizer used and the plant's height might be nonlinear. Initially, adding more fertilizer significantly increases growth, but beyond a certain point, the increase in height diminishes, and the relationship becomes nonlinear. Recognizing this nonlinearity helps optimize fertilizer usage, preventing waste and maximizing plant growth. Similarly, in economics, the relationship between advertising expenditure and sales revenue often exhibits nonlinearity, where initial investments yield high returns, but subsequent investments produce smaller incremental gains.
