To find the profit function \( P(x) \), we need to calculate the difference between the revenue function \( R(x) \) and the cost function \( C(x) \). The given functions are defined as follows:
[tex]\[ C(x) = 400x^2 + 600x \][/tex]
[tex]\[ R(x) = -0.4x^3 + 600x^2 - 200x + 500 \][/tex]
The profit function \( P(x) \) is:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substituting the given functions, we get:
[tex]\[ P(x) = (-0.4x^3 + 600x^2 - 200x + 500) - (400x^2 + 600x) \][/tex]
Distribute the negative sign to the terms in the cost function:
[tex]\[ P(x) = -0.4x^3 + 600x^2 - 200x + 500 - 400x^2 - 600x \][/tex]
Combine like terms:
- The \( x^3 \) term: \(-0.4x^3\)
- The \( x^2 \) terms: \(600x^2 - 400x^2 = 200x^2\)
- The \( x \) terms: \(-200x - 600x = -800x\)
- The constant term: \(500\)
So, the profit function is:
[tex]\[ P(x) = -0.4x^3 + 200x^2 - 800x + 500 \][/tex]
Now, let's verify which option matches our derived profit function:
A. \( P(x) = -0.4x^3 + 1000x^2 + 400x + 500 \)
B. \( P(x) = 0.4x^3 + 200x^2 - 800x + 500 \)
C. \( P(x) = -0.4x^3 + 200x^2 - 800x + 500 \)
D. \( P(x) = 0.4x^3 - 200x^2 + 800x - 500 \)
The correct answer is:
C. [tex]\( P(x) = -0.4x^3 + 200x^2 - 800x + 500 \)[/tex]
