To find the profit function [tex]\( P(x) \)[/tex] of the company, we need to subtract the cost function [tex]\( C(x) \)[/tex] from the revenue function [tex]\( R(x) \)[/tex].
Given revenue function:
[tex]\[ R(x) = 6x^2 + 100x + 300 \][/tex]
Given cost function:
[tex]\[ C(x) = 25x + 100 \][/tex]
The profit function [tex]\( P(x) \)[/tex] is calculated as follows:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute the given functions into the equation:
[tex]\[ P(x) = (6x^2 + 100x + 300) - (25x + 100) \][/tex]
Now, we simplify the expression:
1. Distribute the negative sign through the cost function:
[tex]\[ P(x) = 6x^2 + 100x + 300 - 25x - 100 \][/tex]
2. Combine like terms:
[tex]\[ P(x) = 6x^2 + (100x - 25x) + (300 - 100) \][/tex]
[tex]\[ P(x) = 6x^2 + 75x + 200 \][/tex]
Therefore, the profit function [tex]\( P(x) \)[/tex] is:
[tex]\[ P(x) = 6x^2 + 75x + 200 \][/tex]
The correct answer is:
A. [tex]\( P(x) = 6x^2 + 75x + 200 \)[/tex]
