Question

A company produces x units of product per month, where c(x) represents the total cost and r(x) represents the total revenue for the month. The functions are modeled by c(x) = 425x - 50x² - 30 and r(x) = 300x⁷ - 200x⁷5. The profit, p(x), is the difference between the revenue and the cost where p(x) = r(x) - c(x). What is the profit, p(x)?

Answer 1

The profit function, p(x), is obtained by subtracting the cost function, c(x), from the revenue function, r(x). There appears to be a typo in the revenue function's exponent, but if assuming a correct form of r(x) = 300x², the profit function would be p(x) = 350x² - 425x + 30.

Step-by-step explanation:

To calculate the profit function, p(x), for a company, we subtract the total cost function, c(x), from the total revenue function, r(x). The cost function is given by c(x) = 425x - 50x² - 30 and the revenue function is given by r(x) = 300x⁷ - 200x⁷5. The profit function is therefore p(x) = r(x) - c(x).

By substituting the given functions into p(x), we obtain:

p(x) = (300x⁷ - 200x⁷5) - (425x - 50x² - 30)

However, there appears to be a typo in the revenue function's exponent. Assuming it should be r(x) = 300x² (since 200x⁷5 appears to be incorrect), the correct profit function would then be:

p(x) = (300x²) - (425x - 50x² - 30)

Which simplifies to:

p(x) = 300x² - 425x + 50x² + 30 = 350x² - 425x + 30,

where this final expression represents the firm's profit at different levels of output, x.