To determine the equation of the line of best fit for Aiden's data, we follow these steps:
1. Identify the given data points:
- Arm lengths: [40, 0, 160, 40, 36, 190]
- Corresponding horizontal distances: [1400, 3524, 348, 3, 313, 30]
2. Use the least squares method:
This method helps determine the best fitting line for a set of data points by minimizing the sum of the squares of the offsets (the residuals) of the points from the line. Essentially, we are looking to find the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] of the linear equation [tex]\( y = mx + b \)[/tex].
3. Obtain the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex]:
Based on the calculations, the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] are determined to be:
- [tex]\( m = -10.3 \)[/tex]
- [tex]\( b = 1732.5 \)[/tex]
4. Write the equation of the best fit line:
Substitute the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the linear equation [tex]\( y = mx + b \)[/tex].
So, the equation of the line of best fit for Aiden's data is:
[tex]\[ y = -10.3x + 1732.5 \][/tex]
This equation helps predict the horizontal distance the ball will travel for any given catapult arm length within the tested range.