Question

1. suppose you are studying the growth of a certain plant species over the course of months. you record the following data:

plot the data to visually inspect:
find the first differences and ratios
determine if the data should be modeled linearly or exponentially:

Explanation:

Step1: Calculate first differences

First differences are calculated by subtracting the previous height from the current height.
For month \(1\): \(3 - 1=2\)
For month \(2\): \(9 - 3 = 6\)
For month \(3\): \(27 - 9=18\)
For month \(4\): \(81 - 27 = 54\)
For month \(5\): \(243 - 81=162\)

Step2: Calculate ratios

Ratios are calculated by dividing the current height by the previous height.
For month \(1\): \(\frac{3}{1}=3\)
For month \(2\): \(\frac{9}{3}=3\)
For month \(3\): \(\frac{27}{9}=3\)
For month \(4\): \(\frac{81}{27}=3\)
For month \(5\): \(\frac{243}{81}=3\)

Step3: Determine the model type

Since the first differences are not constant (\(2
eq6
eq18
eq54
eq162\)) and the ratios are constant (\(3 = 3=3=3=3\)), the data follows an exponential model.

Answer:

Year	Height (in)	First Differences	Ratios
1	3	2	3
2	9	6	3
3	27	18	3
4	81	54	3
5	243	162	3

The data should be modeled exponentially.