To find the profit function, [tex]\( P(x) \)[/tex], we need to calculate the difference between the revenue function, [tex]\( S(x) \)[/tex], and the cost function, [tex]\( C(x) \)[/tex].
The given functions are:
[tex]\[ S(x) = 100x^3 + 6x^2 + 97x + 215 \][/tex]
[tex]\[ C(x) = 88x + 215 \][/tex]
The profit function [tex]\( P(x) \)[/tex] is defined as:
[tex]\[ P(x) = S(x) - C(x) \][/tex]
Now, let’s perform the subtraction step-by-step:
1. Write down the revenue function [tex]\( S(x) \)[/tex] and the cost function [tex]\( C(x) \)[/tex]:
[tex]\[ S(x) = 100x^3 + 6x^2 + 97x + 215 \][/tex]
[tex]\[ C(x) = 88x + 215 \][/tex]
2. Subtract [tex]\( C(x) \)[/tex] from [tex]\( S(x) \)[/tex]:
[tex]\[ P(x) = (100x^3 + 6x^2 + 97x + 215) - (88x + 215) \][/tex]
3. Distribute the negative sign and combine like terms:
[tex]\[ P(x) = 100x^3 + 6x^2 + 97x + 215 - 88x - 215 \][/tex]
4. Combine the constants and like terms:
[tex]\[ P(x) = 100x^3 + 6x^2 + (97x - 88x) + (215 - 215) \][/tex]
[tex]\[ P(x) = 100x^3 + 6x^2 + 9x \][/tex]
Therefore, the correct profit function [tex]\( P(x) \)[/tex] is:
[tex]\[ P(x) = 100x^3 + 6x^2 + 9x \][/tex]
The correct answer is:
A. [tex]\( P(x) = 100x^3 + 6x^2 + 9x \)[/tex]
