To determine whether the function describing the total cost of trees purchased is linear or nonlinear, we need to analyze the rate of change in the total cost as the number of trees purchased increases. The rate of change can be determined by calculating the difference in total cost between successive numbers of trees.
Let's calculate the rate of change between each pair of successive tree purchases based on the provided data:
1. Calculate the rate of change between purchasing 1 tree and purchasing 2 trees:
[tex]\[ \Delta y_{1-2} = \text{Cost for 2 trees} - \text{Cost for 1 tree} \][/tex]
[tex]\[ \Delta y_{1-2} = 120 - 60 = 60 \][/tex]
2. Calculate the rate of change between purchasing 2 trees and purchasing 3 trees:
[tex]\[ \Delta y_{2-3} = \text{Cost for 3 trees} - \text{Cost for 2 trees} \][/tex]
[tex]\[ \Delta y_{2-3} = 180 - 120 = 60 \][/tex]
3. Calculate the rate of change between purchasing 3 trees and purchasing 4 trees:
[tex]\[ \Delta y_{3-4} = \text{Cost for 4 trees} - \text{Cost for 3 trees} \][/tex]
[tex]\[ \Delta y_{3-4} = 240 - 180 = 60 \][/tex]
4. Calculate the rate of change between purchasing 4 trees and purchasing 5 trees:
[tex]\[ \Delta y_{4-5} = \text{Cost for 5 trees} - \text{Cost for 4 trees} \][/tex]
[tex]\[ \Delta y_{4-5} = 290 - 240 = 50 \][/tex]
Now that we have the rates of change, we can summarize them:
- Rate of change between 1 and 2 trees: [tex]\( 60 \)[/tex]
- Rate of change between 2 and 3 trees: [tex]\( 60 \)[/tex]
- Rate of change between 3 and 4 trees: [tex]\( 60 \)[/tex]
- Rate of change between 4 and 5 trees: [tex]\( 50 \)[/tex]
In a linear function, the rate of change (or slope) between any two points remains constant. However, we can observe that the rate of change between purchasing 4 and 5 trees ([tex]\( 50 \)[/tex]) is different from the rate of change between purchasing 2 and 3 trees ([tex]\( 60 \)[/tex]). This indicates that the function is nonlinear.
Therefore, the best description of why the function is nonlinear is:
"The rate of change between 2 and 3 trees is different than the rate of change between 3 and 5 trees."
