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In this activity, you will write and solve a system of linear equations given two tables of data. You will then analyze the solution of the system to draw a conclusion.

Part A

While building his catapult, Aiden tested several different designs. He varied the arm length of his catapult and recorded the horizontal distance between where the ball was launched and where it landed for each arm length.

The table below presents Aiden's data showing the length of the catapult arm in relation to the horizontal distance the catapult launched the ball.

| Arm Length (cm) | Distance (m) |
|-----------------|---------------|
| 40 | 1400 |
| 0 | 3524 |
| 160 | 348 |
| 40 | 3 ± 1 |
| 36 | 313 |
| # | 22 |
| ω | 36 |
| 190 | w |
| ss | 30 |
| w | m |

Use the graphing tool to determine the line of best fit for Aiden's data.

Question
What is the equation of the line of best fit for Aiden's data? Enter the correct answer in the form by replacing [tex]\( m \)[/tex] and [tex]\( b \)[/tex] in the equation. Round each number to the nearest tenth.

[tex]\[ y = mx + b \][/tex]

Sagot :

GinnyAnswer
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.
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