Question

the data in the table represents a companys profit based on the number of items produced.

profit based on number of items produced
\

$$\begin{array}{|c|c|} \\hline \\text{items produced } (x) & \\text{dollars of profit } (y) \\\\ \\hline 100 & -70,500 \\\\ \\hline 200 & 50 \\\\ \\hline 300 & 50,100 \\\\ \\hline 400 & 80,300 \\\\ \\hline 500 & 90,400 \\\\ \\hline 600 & 78,000 \\\\ \\hline \\end{array}$$

which equation best represents the data?

\\(y = -1.026x^2 + 1016.402x - 162075\\)
\\(y = -1.036x^2 + 1024.771x - 163710\\)
\\(y = 298.214x - 66317.667\\)
\\(y = 196.2x - 18710\\)

Explanation:

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<reasoning>

Analyze the trend of the data

The data points \((x, y)\) are:

• \((100, -70500)\)
• \((200, 50)\)
• \((300, 50100)\)
• \((400, 80300)\)
• \((500, 90400)\)
• \((600, 78000)\)

The profit \(y\) increases from \(x = 100\) to \(x = 500\) and then decreases at \(x = 600\). This indicates a downward-opening quadratic relationship, so the equation must be of the form \(y = ax^2 + bx + c\) with \(a < 0\).

Test the first quadratic equation

Substitute \(x = 300\) into \(y = -1.026x^2 + 1016.402x - 162075\):
\[

$$\begin{aligned} y &= -1.026(300)^2 + 1016.402(300) - 162075 \\ &= -1.026(90000) + 304920.6 - 162075 \\ &= -92340 + 304920.6 - 162075 \\ &= 50505.6 \end{aligned}$$

\]
This is extremely close to the table value of \(50100\).

Test the second quadratic equation

Substitute \(x = 300\) into \(y = -1.036x^2 + 1024.771x - 163710\):
\[

$$\begin{aligned} y &= -1.036(300)^2 + 1024.771(300) - 163710 \\ &= -1.036(90000) + 307431.3 - 163710 \\ &= -93240 + 307431.3 - 163710 \\ &= 50481.3 \end{aligned}$$

\]
Now substitute \(x = 100\) into both equations to compare:
For the first equation:
\[

$$\begin{aligned} y &= -1.026(100)^2 + 1016.402(100) - 162075 \\ &= -10260 + 101640.2 - 162075 \\ &= -70694.8 \quad (\text{Table value is } -70500) \end{aligned}$$

\]
For the second equation:
\[

$$\begin{aligned} y &= -1.036(100)^2 + 1024.771(100) - 163710 \\ &= -10360 + 102477.1 - 163710 \\ &= -71592.9 \end{aligned}$$

\]
The first equation provides a significantly closer fit to the data.
</reasoning>

<answer>
<mcq-correct>(A) \(y = -1.026x^2 + 1016.402x - 162075\)</mcq-correct>
<mcq-option>(B) \(y = -1.036x^2 + 1024.771x - 163710\)</mcq-option>
<mcq-option>(C) \(y = 298.214x - 66317.667\)</mcq-option>
<mcq-option>(D) \(y = 196.2x - 18710\)</mcq-option>
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Analyze the trend of the data

The data points \((x, y)\) are:

• \((100, -70500)\)
• \((200, 50)\)
• \((300, 50100)\)
• \((400, 80300)\)
• \((500, 90400)\)
• \((600, 78000)\)

The profit \(y\) increases from \(x = 100\) to \(x = 500\) and then decreases at \(x = 600\). This indicates a downward-opening quadratic relationship, so the equation must be of the form \(y = ax^2 + bx + c\) with \(a < 0\).

Test the first quadratic equation

Substitute \(x = 300\) into \(y = -1.026x^2 + 1016.402x - 162075\):
\[

$$\begin{aligned} y &= -1.026(300)^2 + 1016.402(300) - 162075 \\ &= -1.026(90000) + 304920.6 - 162075 \\ &= -92340 + 304920.6 - 162075 \\ &= 50505.6 \end{aligned}$$

\]
This is extremely close to the table value of \(50100\).

Test the second quadratic equation

Substitute \(x = 300\) into \(y = -1.036x^2 + 1024.771x - 163710\):
\[

$$\begin{aligned} y &= -1.036(300)^2 + 1024.771(300) - 163710 \\ &= -1.036(90000) + 307431.3 - 163710 \\ &= -93240 + 307431.3 - 163710 \\ &= 50481.3 \end{aligned}$$

\]
Now substitute \(x = 100\) into both equations to compare:
For the first equation:
\[

$$\begin{aligned} y &= -1.026(100)^2 + 1016.402(100) - 162075 \\ &= -10260 + 101640.2 - 162075 \\ &= -70694.8 \quad (\text{Table value is } -70500) \end{aligned}$$

\]
For the second equation:
\[

$$\begin{aligned} y &= -1.036(100)^2 + 1024.771(100) - 163710 \\ &= -10360 + 102477.1 - 163710 \\ &= -71592.9 \end{aligned}$$

\]
The first equation provides a significantly closer fit to the data.
</reasoning>

<answer>
<mcq-correct>(A) \(y = -1.026x^2 + 1016.402x - 162075\)</mcq-correct>
<mcq-option>(B) \(y = -1.036x^2 + 1024.771x - 163710\)</mcq-option>
<mcq-option>(C) \(y = 298.214x - 66317.667\)</mcq-option>
<mcq-option>(D) \(y = 196.2x - 18710\)</mcq-option>
</answer>

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