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Natalie made a catapult using stronger elastic bands than Allen did. As a result, she was able to send her ball farther with a shorter arm length. She varied the length of her catapult arm while measuring the horizontal distance between where the tennis ball was launched and where it landed. The table presents the data she recorded while using her catapult.

| Arm Length (cm) | Distance (cm) |
|-----------------|---------------|
| 25 | 4011 |
| 35 | 1294 |
| 30 | 1093 |
| 0 | 4 |
| 40 | 4054 |
| 3 | 2011 |
| 4 | 24 |
| 30 | 373 |
| 20 | 338 |

Use the graphing tool to determine the line of best fit for NatalieтАЩs data.

Question: What is the equation of the line of best fit for Natalie's data? Enter the correct answer in the form of [tex]\( y = mx + b \)[/tex], rounding each number to the nearest tenth.



Answer :

GinnyAnswer
To determine the equation of the line of best fit for Natalie's data, follow these steps:

1. Collect the data points:
- Arm lengths (in cm): [tex]\[25, 35, 30, 0, 4, 3, 4, None, 20, None\][/tex]
- Distances (in cm): [tex]\[4011, 1294, None, 4, 4054, 2011, 24, None, 373, 338\][/tex]

2. Exclude missing data:
- Only include pairs where both arm length and distance are present.
- Valid pairs are:
[tex]\[ \begin{align*} (25, 4011), \\ (35, 1294), \\ (0, 4), \\ (4, 4054), \\ (3, 2011), \\ (4, 24), \\ (20, 373). \end{align*} \][/tex]

3. Calculate the line of best fit:
- Using linear regression, determine the slope (m) and intercept (b) for the line. After performing these calculations, you get:
[tex]\[ \begin{align*} \text{slope} (m) &= 20.168772563176898, \\ \text{intercept} (b) &= 1419.377385250129. \end{align*} \][/tex]

4. Form the equation of the line:
- The equation of the line of best fit is in the form [tex]\( y = mx + b \)[/tex].
- Plug in the calculated values for [tex]\(m\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ y = 20.168772563176898x + 1419.377385250129. \][/tex]

5. Round the numbers to the nearest tenth:
- Slope ([tex]\(m\)[/tex]) rounded to the nearest tenth is [tex]\(20.2\)[/tex].
- Intercept ([tex]\(b\)[/tex]) rounded to the nearest tenth is [tex]\(1419.4\)[/tex].

6. Write the final equation:
[tex]\[ y = 20.2x + 1419.4. \][/tex]

Therefore, the equation of the line of best fit for Natalie's data is [tex]\( y = 20.2x + 1419.4 \)[/tex].

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