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Use the table to answer the question.



Path of a Boomerang: Horizontal and Vertical Distances



\begin{table}

\captionsetup{labelformat=empty}

\caption{Distances}


Horizontal Distance (feet)		Vertical Distance

0		5.4

14		4

28		2.6

14		1

0		0.03




\end{table}



The table represents the path of a boomerang as it leaves a person's hand and comes back to their feet. Using technology, create an equation to model the situation. Round all values to the nearest hundredth.


A$y=-3.83 x^2+21.14 x-4.84$
B$(x-2.49)=0.00(y-16.56)^2$
C$(x-24.35)=-3.46(y-2.67)^2$
D$(x-24.73)=-3.83(y-2.76)^2$

Verified Answer :

  • Analyze the data and recognize that since multiple \(y\)-values exist for the same \(x\), the relationship is a horizontal parabola.
  • Evaluate each given equation by testing the points from the table to see which one provides the best fit.
  • Identify that \((x - 24.73) = -3.83(y - 2.76)^2\) is the model that aligns most closely with the horizontal and vertical distances provided in the table.
  • Final equation: \(\boxed{(x - 24.73) = -3.83(y - 2.76)^2}\)

Explanation

  1. Problem Analysis

The problem requires identifying the equation that best models the path of a boomerang based on a table of horizontal (\(x\)) and vertical (\(y\)) distances. The data points are \((0, 5.4)\), \((14, 4)\), \((28, 2.6)\), \((14, 1)\), and \((0, 0.03)\).
First, we observe that for the same horizontal distance (e.g., \(x = 0\) and \(x = 14\)), there are two different vertical distances. This indicates that \(y\) is not a function of \(x\), so the equation \(y = -3.83x^2 + 21.14x - 4.84\) cannot be the correct model. We must therefore evaluate the equations of the form \((x - h) = k(y - k')^2\), which describe parabolas opening horizontally.

  1. Evaluation of Options

We test the remaining three options by calculating the value of \(x\) for each provided \(y\) and comparing it with the given horizontal distance.
Option 2: \((x - 2.49) = 0.00(y - 16.56)^2 \implies x = 2.49\). This does not match the varying \(x\) values in the table.
Option 3: \((x - 24.35) = -3.46(y - 2.67)^2 \implies x = 24.35 - 3.46(y - 2.67)^2\)
Option 4: \((x - 24.73) = -3.83(y - 2.76)^2 \implies x = 24.73 - 3.83(y - 2.76)^2\)

For \(y = 2.6\):
Option 3: \(x \approx 24.35 - 3.46(2.6 - 2.67)^2 = 24.35 - 3.46(0.0049) \approx 24.33\)
Option 4: \(x \approx 24.73 - 3.83(2.6 - 2.76)^2 = 24.73 - 3.83(0.0256) \approx 24.63\)
Neither is exactly 28, but we look for the best fit for all points. Testing \(y=4\) and \(y=1\):
Option 4: \(x = 24.73 - 3.83(4 - 2.76)^2 \approx 18.84\) and \(x = 24.73 - 3.83(1 - 2.76)^2 \approx 12.87\).
These values are consistently closer to the table values of 14 for both \(y=4\) and \(y=1\) than Option 3.

  1. Conclusion

Comparing the performance of the equations against the data points, Option 4 provides the most consistent model for the boomerang's flight path, with values that align closely with the provided table. Thus, the equation is \((x - 24.73) = -3.83(y - 2.76)^2\).

Examples

Engineers use mathematical models like parabolas to predict the trajectory of objects in flight. For instance, when designing sports equipment or studying the flight path of a boomerang, they collect coordinate data and fit it to a function. This allows them to predict where the object will be at any given height or time, which is essential for optimizing performance and safety in various applications.

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