Question

the table gives the number of branches on a large tree after the year 2000. which kind of function best models this relationship?
time (years) \tbranches
0 \t16
2 \t23
4 \t33
6 \t48
8 \t69
10 \t99
choose 1 answer:
a linear
b exponential

Explanation:

Step1: Check linear model (constant difference)

Calculate differences between consecutive branch counts:
$23-16=7$, $33-23=10$, $48-33=15$, $69-48=21$, $99-69=30$

Step2: Check exponential model (constant ratio)

Calculate ratios between consecutive branch counts:
$\frac{23}{16}=1.4375$, $\frac{33}{23}\approx1.4348$, $\frac{48}{33}\approx1.4545$, $\frac{69}{48}=1.4375$, $\frac{99}{69}\approx1.4348$

Step3: Compare consistency

Differences increase, ratios stay nearly constant.

Answer:

B. Exponential