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

Total daily cost = C = 45x +27 where x is the number of units sold or produced.

Price/unit = P = 60 - 0.5x

Revenue/day = R = Px = x(60 - 0.5x) = 60x - 0.5x²

Profit = R - C = 60x - 0.5x² - (45x + 27) -> 60x - 0.5x² - 45x - 27 -> -0.5x² + 15x - 27.

The "a" is neg. so the parabola opens downwards and there is a maximum at x = -b/2a

-b/2a = -15/-1 = 15 units

When x = 15, Profit = -0.5(15²) + 15(15) - 27 -> -112.5 + 225 - 27 = $85.5 (try the equation on the Desmos Graphing Calculator site to confirm).

Could also take the 1st derivative of the profit expression and set it = to 0 to get x value at maximum profit.

Answer 2

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) = $85.5 per day}