Bot GPT 4-o mini To determine which of the model equations correctly represents the trajectory of the boomerang, we need to analyze the data provided and estimate an appropriate quadratic model or check given options for plausibility.
Based on the points in the table, the trajectory of the boomerang likely follows a parabolic arc, which can be modeled by a quadratic equation in the form: \[ y = ax^2 + bx + c \]
Let's analyze the trends in the given data:
- As the horizontal distance (x) increases initially from 0 to 28 feet, the vertical distance (y) decreases.
- After reaching the minimum vertical distance, it then moves back up towards 0 feet as it returns.
The proper equation would likely have a negative "a" value due to the downward opening of a parabola.
Knowing this, we can assess the options given:
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\((x−2.49) = 0.00(y−16.56)^2\) - This suggests a vertical line, which isn't suitable for this data set.
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\((x−24.35) = -3.46(y−2.67)^2\) - This does describe a parabola that opens to the left, not matching the left-to-right configuration needed.
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\(y = -3.83x^2 + 21.14x - 4.84\) - This is a downward-opening parabola which fits our expected shape.
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\((x−24.73) = -3.83(y−2.76)^2\) - Similar to option 2, this equation also describes a leftward-opening parabola.
Given the analysis, the most appropriate equation that models the boomerang's trajectory in this case is:
\(y = -3.83x^2 + 21.14x - 4.84\)
This provides a suitable quadratic model responsive to the data given and aligns with the observed trajectory of the boomerang.
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