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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 show the data for two of the trees.

\begin{tabular}{|r|c|}
\multicolumn{2}{c}{Tree 1} \\
\hline Year & \begin{tabular}{c}
Trunk \\
Diameter \\
(inches)
\end{tabular} \\
\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{tabular}

\begin{tabular}{|r|c|}
\multicolumn{2}{c}{Tree 2} \\
\hline Year & \begin{tabular}{c}
Trunk \\
Diameter \\
(inches)
\end{tabular} \\
\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{tabular}

The scientist creates an equation that models the data for each tree so that she can predict the trunk 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.

Equation for Tree 1:
[tex]\[
y = \square + \square x
\][/tex]

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

Sagot :

GinnyAnsweridn
Let's go through the steps to complete the linear equation that models the data for Tree 1.

We have the data for 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]

To find the linear equation [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, and
- [tex]\( b \)[/tex] is the y-intercept.

From our calculations, we have:
- The slope [tex]\( m \)[/tex] = 0.3
- The y-intercept [tex]\( b \)[/tex] = 18.3

Hence, the linear equation is:

[tex]\[ y = 0.3x + 18.3 \][/tex]

So, the completed equation for Tree 1 is:

[tex]\[ y = 0.3x + 18.3 \][/tex]

This equation can be used to predict the trunk diameter in the future based on the year.
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