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]
