Thematic Reasoning

Question 6

A scientist is studying red maple tree growth in a state. She measured the trunk diameters of a sample of trees in the same month every other year. The tables below show the data for two of the trees.

Tree 1
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Year} & \text{Trunk diameter (inches)} \\
\hline
1 & 18.6 \\
\hline
3 & 19.2 \\
\hline
5 & 19.8 \\
\hline
7 & 20.4 \\
\hline
9 & 21.0 \\
\hline
11 & 21.6 \\
\hline
13 & 22.2 \\
\hline
\end{array}
\][/tex]

Tree 2
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Year} & \text{Trunk diameter (inches)} \\
\hline
1 & 11.4 \\
\hline
3 & 12.0 \\
\hline
5 & 12.6 \\
\hline
7 & 13.2 \\
\hline
9 & 13.8 \\
\hline
11 & 14.4 \\
\hline
13 & 15.0 \\
\hline
\end{array}
\][/tex]

This is the final year in which she will collect data. When her data collection is complete, she will predict future red maple tree growth.

The scientist creates an equation that models her data for each tree so that she can predict the diameter in the future. Complete a linear equation that fits the data for Tree 1, where [tex]$x$[/tex] is the year and [tex]$y$[/tex] is the trunk diameter in inches. Click on the variables and numbers you want to select and drag them into the boxes.

Equation for Tree 1

[tex]\[
\begin{array}{l}
y = \square + \square x \\
-0.6 \quad -0.3 \quad 0.3 \quad 0.6 \\
18.0 \quad 18.3 \quad 18.6
\end{array}
\][/tex]

Choices:
[tex]\[
-0.3 \quad 0.3 \quad 0.6 \quad 18.0 \quad 18.3 \quad 18.6
\][/tex]



Answer :

The scientist is studying the growth of a red maple tree over several years and has collected measurements to create a linear model of the tree's trunk diameter over time. Let's create a linear equation for tree 1 based on the collected data.

### Data for Tree 1

| Year | Trunk Diameter (inches) |
|------|-------------------------|
| 1 | 18.6 |
| 3 | 19.2 |
| 5 | 19.8 |
| 7 | 20.4 |
| 9 | 21.0 |
| 11 | 21.6 |
| 13 | 22.2 |

The linear equation that models this data can be written in the form:
[tex]\[ y = mx + b \][/tex]
where
- [tex]\( y \)[/tex] is the trunk diameter,
- [tex]\( x \)[/tex] is the year,
- [tex]\( m \)[/tex] is the slope of the line (i.e., the rate of growth per year),
- [tex]\( b \)[/tex] is the y-intercept (i.e., the trunk diameter at year 0).

From the provided data, we obtain the slope ([tex]\( m \)[/tex]) and the intercept ([tex]\( b \)[/tex]) of the linear model.

With the slope being [tex]\( 0.3 \)[/tex] and the intercept being [tex]\( 18.3 \)[/tex], the equation becomes:
[tex]\[ y = 0.3x + 18.3 \][/tex]

Thus, the completed linear equation that fits the data for tree 1 is:
[tex]\[ y = 0.3x + 18.3 \][/tex]

Here are the elements you can place in the equation:
- [tex]\( y \)[/tex]
- [tex]\( = \)[/tex]
- [tex]\( 0.3 \)[/tex]
- [tex]\( x \)[/tex]
- [tex]\( + \)[/tex]
- [tex]\( 18.3 \)[/tex]

The complete equation for tree 1 is:
[tex]\[ y = 0.3x + 18.3 \][/tex]

This equation enables the scientist to predict the diameter of the tree trunk in future years based on the linear growth model.