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Find the profit function \( P(x) \) for a corporation for the years 2004−2012, given the revenue function \( R(x) \) and the cost function \( C(x) \) for the years 2004−2012.

Given:
\[ R(x) = -(0.0685x^3) + (1.804x^2) - 10.27x + 119.5 \]
\[ C(x) = -(0.0194x^3) + (0.966x^2) - 7.51x + 97.8 \]

Calculate:
\[ P(x) = R(x) - C(x) \]

(Simplify your answer.)

Answer :

Final answer:

The profit function P(x) is computed as the difference between the revenue function R(x) and the costs function C(x). By substantifying these formulas into the profit function formula, we get P(x) = -0.0491x^3 + 0.838x² - 2.76x + 21.7

Explanation:

The profit function P(x) could be obtained from the revenue function, R(x) and the costs function, C(x). The profit for a corporation is calculated as the difference between the revenue (the money made from selling goods or services) and the costs (the money expended in producing goods or services).

Thus, to get the profit function, you subtract the cost function from the revenue function.

Therefore, P(x) = R(x) - C(x)

Substituting the given functions into the formula, we get:

P(x) = [-(0.0685x^3) +(1.804x²) −10.27x+119.5] - [-(0.0194x^3) +(0.966x²) −7.51x+97.8]

This simplifies to: P(x) = -(0.0685-0.0194) x^3 + (1.804-0.966) x² - (10.27-7.51) x + (119.5-97.8)

Therefore, the profit function is P(x) = -0.0491x^3 + 0.838x² - 2.76x + 21.7

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