To determine whether the relationship in the table is linear or nonlinear, we need to examine the changes in the "Total Houses Built" compared to the changes in the "Months Passed" and see if the rate of change is constant.
The data provided in the table is as follows:
| Months Passed | Total Houses Built |
|---------------|---------------------|
| 0 | 0 |
| 3 | 33 |
| 4 | 46 |
| 8 | 108 |
Let's go through a detailed, step-by-step analysis:
1. Calculate the differences in months (Δ months):
- From month 0 to month 3: [tex]\( 3 - 0 = 3 \)[/tex]
- From month 3 to month 4: [tex]\( 4 - 3 = 1 \)[/tex]
- From month 4 to month 8: [tex]\( 8 - 4 = 4 \)[/tex]
So, the differences in months are: [tex]\( [3, 1, 4] \)[/tex].
2. Calculate the differences in total houses built (Δ houses):
- From 0 houses to 33 houses: [tex]\( 33 - 0 = 33 \)[/tex]
- From 33 houses to 46 houses: [tex]\( 46 - 33 = 13 \)[/tex]
- From 46 houses to 108 houses: [tex]\( 108 - 46 = 62 \)[/tex]
So, the differences in total houses built are: [tex]\( [33, 13, 62] \)[/tex].
3. Calculate the rate of change for each interval:
- From month 0 to 3: [tex]\( \frac{33}{3} = 11.0 \)[/tex] houses per month
- From month 3 to 4: [tex]\( \frac{13}{1} = 13.0 \)[/tex] houses per month
- From month 4 to 8: [tex]\( \frac{62}{4} = 15.5 \)[/tex] houses per month
So, the rates of change are: [tex]\( [11.0, 13.0, 15.5] \)[/tex].
4. Check if the rate of change is constant:
- The rate of change from 0 to 3 months is [tex]\( 11.0 \)[/tex]
- The rate of change from 3 to 4 months is [tex]\( 13.0 \)[/tex]
- The rate of change from 4 to 8 months is [tex]\( 15.5 \)[/tex]
The rates of change are [tex]\( 11.0 \)[/tex], [tex]\( 13.0 \)[/tex], and [tex]\( 15.5 \)[/tex]. Since these rates are not equal, we can conclude that the rate of change is not constant.
Based on the analysis, the relationship between the "Total Houses Built" and the "Months Passed" does not show a constant rate of change. Therefore, the data set is nonlinear.
So, the best description of the data set 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.
