To find the equation representing the profit [tex]\( P \)[/tex] for the business, we start by understanding that profit is the difference between revenue [tex]\( R \)[/tex] and cost [tex]\( C \)[/tex]. Therefore, we need to subtract the cost function from the revenue function.
Given:
- The revenue function [tex]\( R(x) = x^3 - 6x^2 + x + 75 \)[/tex]
- The cost function [tex]\( C(x) = x^2 - 70x \)[/tex]
The profit function [tex]\( P(x) \)[/tex] is derived by subtracting the cost function from the revenue function:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute the given functions:
[tex]\[ P(x) = (x^3 - 6x^2 + x + 75) - (x^2 - 70x) \][/tex]
Now, distribute and combine like terms:
[tex]\[ P(x) = x^3 - 6x^2 + x + 75 - x^2 + 70x \][/tex]
Simplify by combining the [tex]\( x^2 \)[/tex] terms and the [tex]\( x \)[/tex] terms:
[tex]\[ P(x) = x^3 - 6x^2 - x^2 + x + 70x + 75 \][/tex]
[tex]\[ P(x) = x^3 - 7x^2 + 71x + 75 \][/tex]
Thus, the equation that represents the profit [tex]\( P \)[/tex] for this business is:
[tex]\[ P = x^3 - 7x^2 + 71x + 75 \][/tex]
Among the given options, the correct one is:
[tex]\[ P = x^3 - 7x^2 + 71x + 75 \][/tex]
Therefore, the answer is:
[tex]\[ P = x^3 - 7x^2 + 71x + 75 \][/tex]
