Maximize Profit 3E97Ed

1. **State the problem:** We are given the cost function $$C(x) = x^3 - 820.5x^2 - 31587x + 1082$$ and the demand function $$p(x) = 34.5x + 165.$$ We need to find the production level $$x$$ that maximizes profit. 2. **Recall the profit function:** Profit $$P(x)$$ is revenue minus cost. $$P(x) = R(x) - C(x)$$ where revenue $$R(x) = x \cdot p(x)$$. 3. **Calculate revenue function:** $$R(x) = x \times (34.5x + 165) = 34.5x^2 + 165x$$ 4. **Write profit function:** $$P(x) = R(x) - C(x) = (34.5x^2 + 165x) - (x^3 - 820.5x^2 - 31587x + 1082)$$ Simplify: $$P(x) = 34.5x^2 + 165x - x^3 + 820.5x^2 + 31587x - 1082$$ $$P(x) = -x^3 + (34.5 + 820.5)x^2 + (165 + 31587)x - 1082$$ $$P(x) = -x^3 + 855x^2 + 31752x - 1082$$ 5. **Find critical points by differentiating profit:** $$P'(x) = \frac{d}{dx}(-x^3 + 855x^2 + 31752x - 1082) = -3x^2 + 1710x + 31752$$ 6. **Set derivative equal to zero to find critical points:** $$-3x^2 + 1710x + 31752 = 0$$ Divide both sides by -3: $$\cancel{-3}x^2 + \cancel{-3} \times (-570)x + \cancel{-3} \times (-10584) = 0$$ $$x^2 - 570x - 10584 = 0$$ 7. **Solve quadratic equation:** Use quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=-570$$, $$c=-10584$$. Calculate discriminant: $$\Delta = (-570)^2 - 4(1)(-10584) = 324900 + 42336 = 367236$$ Calculate square root: $$\sqrt{367236} \approx 606.01$$ Calculate roots: $$x_1 = \frac{570 + 606.01}{2} = \frac{1176.01}{2} = 588.005$$ $$x_2 = \frac{570 - 606.01}{2} = \frac{-36.01}{2} = -18.005$$ 8. **Interpret roots:** Since production level $$x$$ cannot be negative, discard $$x_2 = -18.005$$. 9. **Check second derivative to confirm maximum:** $$P''(x) = \frac{d}{dx}P'(x) = \frac{d}{dx}(-3x^2 + 1710x + 31752) = -6x + 1710$$ Evaluate at $$x = 588.005$$: $$P''(588.005) = -6(588.005) + 1710 = -3528.03 + 1710 = -1818.03 < 0$$ Since $$P''(x) < 0$$, this critical point is a local maximum. 10. **Final answer:** The production level that maximizes profit is approximately $$588$$ (rounded to the nearest integer).