1. **Problem Statement:** Given the cost function $C(x) = 24x + 21900$ and revenue function $R(x) = 200x - 0.2x^2$ for ski jackets, we need to: i. Find the profit function $G(x)$. ii. Find the breakeven points and interpret them. iii. Find the marginal profit function and evaluate it at $x=900$ units. 2. **Formulas and Rules:** - Profit function: $G(x) = R(x) - C(x)$. - Breakeven points occur where profit is zero: $G(x) = 0$. - Marginal profit is the derivative of the profit function: $G'(x)$. 3. **Step i: Find the profit function $G(x)$** $$G(x) = R(x) - C(x) = (200x - 0.2x^2) - (24x + 21900)$$ Simplify: $$G(x) = 200x - 0.2x^2 - 24x - 21900 = (200x - 24x) - 0.2x^2 - 21900 = 176x - 0.2x^2 - 21900$$ So, $$G(x) = -0.2x^2 + 176x - 21900$$ 4. **Step ii: Find breakeven points where $G(x) = 0$** Set profit to zero: $$-0.2x^2 + 176x - 21900 = 0$$ Multiply both sides by $-5$ to clear decimals: $$\cancel{-0.2} \times -5 x^2 + \cancel{176} \times -5 x - 21900 \times -5 = 0 \times -5$$ $$x^2 - 880x + 109500 = 0$$ Use quadratic formula: $$x = \frac{880 \pm \sqrt{880^2 - 4 \times 1 \times 109500}}{2}$$ Calculate discriminant: $$880^2 = 774400$$ $$4 \times 109500 = 438000$$ $$\sqrt{774400 - 438000} = \sqrt{336400} = 580$$ So, $$x = \frac{880 \pm 580}{2}$$ Two solutions: $$x_1 = \frac{880 - 580}{2} = \frac{300}{2} = 150$$ $$x_2 = \frac{880 + 580}{2} = \frac{1460}{2} = 730$$ **Interpretation:** Breakeven points at 150 and 730 jackets mean the company neither makes profit nor loss at these production levels. Between these points, profit is positive; outside, it is negative. 5. **Step iii: Marginal profit function and evaluation at $x=900$** Marginal profit is derivative of $G(x)$: $$G'(x) = \frac{d}{dx}(-0.2x^2 + 176x - 21900) = -0.4x + 176$$ Evaluate at $x=900$: $$G'(900) = -0.4 \times 900 + 176 = -360 + 176 = -184$$ **Interpretation:** At 900 units, marginal profit is -184, meaning producing one more jacket decreases profit by 184 dollars. Production beyond 730 units reduces profit. --- **Summary:** - Profit function: $G(x) = -0.2x^2 + 176x - 21900$ - Breakeven points: $x=150$ and $x=730$ - Marginal profit: $G'(x) = -0.4x + 176$ - Marginal profit at 900 units: $-184$ (loss per additional unit)