Maximum Possible Profit

Question

The total cost of producing x items per day is (45x+27)dollars , and the price per item at which each may be sold is (60-0.5x)dollars . find an expression for the daily profit, and hence find the maximum possible profit?

Answer 1

Revenu R(x) = x(60 - 0.5x);

Profit P(x) = R(x) - C(x) = 60x - 0.5x² - 45x - 27; P(x) = 15x - 0.5x² - 27.

P(x) = - 0.5(x² - 30x + 225) + 112.5 - 27 = 85.5 - 0.5(x - 15)²

^{So, max P(x) = P(15) = $88.5 per day}