The data in the table represents a company's profit based on the number of items produced.

Profit Based on Number of Items Produced

\begin{tabular}{|c|c|}
\hline
Items produced [tex]$(x)$[/tex] & Dollars of profit [tex]$(y)$[/tex] \\
\hline
100 & [tex]$-70,500$[/tex] \\
\hline
200 & 50 \\
\hline
300 & 50,100 \\
\hline
400 & 80,300 \\
\hline
500 & 90,400 \\
\hline
600 & 78,000 \\
\hline
\end{tabular}

Which equation best represents the data?

A. [tex]$y = -1.026 x^2 + 1016.402 x - 162075$[/tex]

B. [tex]$y = -1.036 x^2 + 1024.771 x - 163710$[/tex]

C. [tex]$y = 298.214 x - 66317.667$[/tex]

D. [tex]$y = 196.2 x - 18710$[/tex]



Answer :

To determine which equation best represents the data, we will analyze the profit values based on the number of items produced.

Let's review the equations provided:

1. [tex]\( y = -1.026x^2 + 1016.402x - 162075 \)[/tex]
2. [tex]\( y = -1.036x^2 + 1024.771x - 163710 \)[/tex]
3. [tex]\( y = 298.214x - 66317.667 \)[/tex]
4. [tex]\( y = 196.2x - 18710 \)[/tex]

We will see the general behavior of the given data:
- At [tex]\( x = 100 \)[/tex], [tex]\( y = -70,500 \)[/tex]
- At [tex]\( x = 200 \)[/tex], [tex]\( y = 50 \)[/tex]
- At [tex]\( x = 300 \)[/tex], [tex]\( y = 50,100 \)[/tex]
- At [tex]\( x = 400 \)[/tex], [tex]\( y = 80,300 \)[/tex]
- At [tex]\( x = 500 \)[/tex], [tex]\( y = 90,400 \)[/tex]
- At [tex]\( x = 600 \)[/tex], [tex]\( y = 78,000 \)[/tex]

This spread of data suggests a non-linear relationship, possibly quadratic given the profit values peak and then decrease.

Selecting the best-fit equation:
1. [tex]\( y = -1.026x^2 + 1016.402x - 162075 \)[/tex]
2. [tex]\( y = -1.036x^2 + 1024.771x - 163710 \)[/tex]
3. [tex]\( y = 298.214x - 66317.667 \)[/tex]
4. [tex]\( y = 196.2x - 18710 \)[/tex]

Considering the nature of the coefficients and the polynomial degree, as well as confirming the given consistent answer:
[tex]\[ \boxed{\text{None}} \][/tex]