a. [tex]\boxed{y-602=\frac{2}{3}(x-3)}[/tex]
b. For every month the pine tree grows about 0.67 inches.
c. It represents the height of the tree at the moment Ariel started to record the heights.
Explanation:
PART A.
In this exercise we have that in July Ariel recorded the height of a pine tree and how quickly it was expected to grow in next several months. From the Table, we can get the following points:
[tex]P_{1}(3,602) \\ \\ P_{2}(6,604) \\ \\ P_{3}(q,606) \\ \\ P_{4}(12,608)[/tex]
It is obvious that this problem follows a linear equation, so our goal is to find the equation that matches the Table.
The point slope form of the equation of a line is:
[tex]y-y_{1}=m(x-x_{1}) \\ \\ Where: \\ \\ m:Slope \\ \\ (x_{1}, y_{1}):A \ point \ on \ the \ line[/tex]
Finding (m):
[tex]Using: \\ \\ P_{1}(x_{1},y_{1}) \rightarrow P_{1}(3,602) \\ \\ P_{2}(x_{2}, y_{2}) \rightarrow P_{2}(6,604) \\ \\ m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{604-602}{6-3} \\ \\ m=\frac{2}{3}[/tex]
Finally, the equation of the line is:
[tex](x_{1}, y_{1}) \rightarrow (3,602) \\ \\ \boxed{y-602=\frac{2}{3}(x-3)}[/tex]
PART B.
In this case, we have that:
- The x-axis represents the Number of Months the pine tree was expected to grow.
- The y-axis represents the Height of the Tree in inches.
Since the slope o a line is:
[tex]Slope=\frac{Change \ in \ y}{Change \ in \ x}=\frac{2}{3}[/tex]
Then, this means that every three months the pine tree grows two inches, or for every month the pine tree grows about 0.67 inches.
PART C.
The y-intercept can be found setting [tex]x=0[/tex]. So, from our equation:
[tex]y-602=\frac{2}{3}(x-3)} \\ \\ \\ If \ x=0: \\ \\ y-602=\frac{2}{3}(0-3) \\ \\ y-602=-2 \\ \\ Solving \ for \ y: \\ \\ y=602-2 \\ \\ y=600[/tex]
So the y-intercept is [tex]y=600[/tex] and represents the height of the tree at the moment Ariel started to record the heights.
Learn more:
Slope Intercept form: https://brainly.com/question/4192440
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