To determine whether the relationship between time and the number of branches on the tree is best modeled by a linear or exponential function, we need to analyze how the number of branches changes over time. Let's look step-by-step at two different possible growth patterns: linear and exponential.
1. Linear Growth Analysis:
In linear growth, the difference between successive values should be approximately constant. We can calculate the differences between the number of branches at successive time intervals.
- Difference between year 2 and year 0: [tex]\(23 - 16 = 7\)[/tex]
- Difference between year 4 and year 2: [tex]\(33 - 23 = 10\)[/tex]
- Difference between year 6 and year 4: [tex]\(48 - 33 = 15\)[/tex]
- Difference between year 8 and year 6: [tex]\(69 - 48 = 21\)[/tex]
- Difference between year 10 and year 8: [tex]\(99 - 69 = 30\)[/tex]
The differences are: [tex]\(7, 10, 15, 21, 30\)[/tex]. These differences are not constant, indicating that the growth is not linear.
2. Exponential Growth Analysis:
In exponential growth, the ratio between successive values should be approximately constant. We can calculate the ratios between the number of branches at successive time intervals.
- Ratio between year 2 and year 0: [tex]\( \frac{23}{16} \approx 1.4375 \)[/tex]
- Ratio between year 4 and year 2: [tex]\( \frac{33}{23} \approx 1.4348 \)[/tex]
- Ratio between year 6 and year 4: [tex]\( \frac{48}{33} \approx 1.4545 \)[/tex]
- Ratio between year 8 and year 6: [tex]\( \frac{69}{48} \approx 1.4375 \)[/tex]
- Ratio between year 10 and year 8: [tex]\( \frac{99}{69} \approx 1.4348 \)[/tex]
The ratios are approximately close to each other: [tex]\(1.4375, 1.4348, 1.4545, 1.4375, 1.4348\)[/tex]. However, on closer inspection, these ratios are not perfectly consistent.
Based on these calculations:
- The differences between the number of branches do not show a consistent pattern needed for linear growth.
- The ratios between the number of branches are somewhat close to each other but not perfectly consistent.
Given that the growth is neither strictly exponential nor strictly linear, the most appropriate model considering the given inconsistencies is:
Linear.
Therefore, the kind of function that best models this relationship is a linear function. The correct choice is:
(A) Linear
