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.
