Question

graphing practice
practice problem #1
background: the thickness of the annual rings indicates what type of environmental situation was occurring the time of the trees development. a thin ring usually indicates a rough period of development such as lack of water, forest fires, or insect infestation. a thick ring means a prosperous period of development. use the information from the data table below to create a proper scientific graph and to answer the corresponding questions
age of trees (in years) average thickness of annual rings in forest a (millimeters) average thickness of annual rings in forest b (millimeters)
10 20 24
20 24 35
30 30 38
35 34 45
50 41 51
60 46

1. what is the dependent variable?
2. what is the independent variable?
3. what was the average thickness of annual rings for 40 - year - old trees in forest a?
4. what is it called when you make predictions within given data, such as made in question #3?
5. what was the mean thickness of annual rings for all trees found in forest b?
6. based on the data shown, what can be concluded about the comparative health of forest a & b?
7. what type of relationship (constant, direct, or indirect) exists between the age of trees and the average thickness of the trees rings? explain.

Explanation:

Step1: Identify dependent variable

The variable that depends on another is the average thickness of annual rings.

Step2: Identify independent variable

The variable that is manipulated or changed is the age of trees.

Step3: Interpolate for 40 - year - old trees in Forest A

For Forest A, the values for 35 - year - old trees is 34 mm and for 50 - year - old trees is 41 mm. Using linear interpolation: Let \(x_1 = 35\), \(y_1=34\), \(x_2 = 50\), \(y_2 = 41\), and \(x = 40\). The formula for linear interpolation is \(y=y_1+\frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}\). Substituting values: \(y = 34+\frac{(40 - 35)(41 - 34)}{50 - 35}=34+\frac{5\times7}{15}=34+\frac{7}{3}\approx36.33\) mm.

Step4: Define prediction within data

It is called interpolation.

Step5: Calculate mean thickness for Forest B

Sum of thickness values for Forest B: \(24 + 28+35+38+45+51 = 221\). There are 6 data - points. Mean \(=\frac{221}{6}\approx36.83\) mm.

Step6: Analyze forest health

Trees in Forest B generally have thicker annual rings, indicating potentially better health or more favorable growing conditions.

Step7: Determine relationship type

As the age of trees increases, the average thickness of the tree's rings generally increases, so it is a direct relationship.

Answer:

1. Average thickness of annual rings
2. Age of trees
3. 36.33 mm
4. Interpolation
5. 36.83 mm
6. Trees in Forest B have thicker annual rings, suggesting better health or growing conditions.
7. Direct relationship. As the age of trees increases, the average thickness of tree rings generally increases.