To determine the equation of the line of best fit for Natalie's data, we analyze the relationship between the lengths of the catapult arm and the corresponding horizontal distances achieved. This relationship can be represented by a linear equation of the form:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\(y\)[/tex] is the horizontal distance,
- [tex]\(x\)[/tex] is the length of the catapult arm,
- [tex]\(m\)[/tex] is the slope of the line,
- [tex]\(b\)[/tex] is the y-intercept.
Given the data:
[tex]\[
\begin{array}{|cc|}
\hline
\text{Length of Catapult Arm} \ ( \text{cm} ) & \text{Horizontal Distance} \ ( \text{cm} ) \\
\hline
25 & 290.8 \\
35 & 325.4 \\
30 & 315.2 \\
60 & 420 \\
65 & 435.8 \\
50 & 385.1 \\
45 & 355 \\
40 & 362 \\
50 & 378.3 \\
40 & 333.9 \\
\hline
\end{array}
\][/tex]
After calculating the line of best fit, we find that the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex] are determined. The slope [tex]\(m\)[/tex] represents the rate of change of the horizontal distance with respect to the catapult arm length, and the y-intercept [tex]\(b\)[/tex] represents the horizontal distance when the catapult arm length is zero (although it is an extrapolated value in this context, as zero arm length is not practically useful).
For this particular data, the values are:
- Slope ([tex]\(m\)[/tex]): 3.6 (rounded to the nearest tenth)
- Intercept ([tex]\(b\)[/tex]): 202.6 (rounded to the nearest tenth)
Thus, the equation of the line of best fit for Natalie's data is:
[tex]\[ y = 3.6x + 202.6 \][/tex]
This equation represents the relationship between the length of Natalie's catapult arm and the horizontal distance the ball travels.
