1. **Problem Statement:** A rectangle is inscribed in a semicircle of radius 2. We want to find the largest possible area of this rectangle and its dimensions. 2. **Setup and Formula:** The semicircle is described by the equation $$x^2 + y^2 = 4$$ with radius 2. The rectangle's width is $$2x$$ (from $$-x$$ to $$x$$ on the x-axis) and height is $$y = \sqrt{4 - x^2}$$. The area function is: $$A(x) = 2x \times \sqrt{4 - x^2}$$ where $$0 \leq x \leq 2$$. 3. **Find the derivative:** Using the product and chain rules: $$A'(x) = 2 \sqrt{4 - x^2} + 2x \times \frac{d}{dx}(\sqrt{4 - x^2})$$ Calculate derivative inside: $$\frac{d}{dx}(\sqrt{4 - x^2}) = \frac{1}{2\sqrt{4 - x^2}} \times (-2x) = \frac{-x}{\sqrt{4 - x^2}}$$ So, $$A'(x) = 2 \sqrt{4 - x^2} + 2x \times \left(-\frac{x}{\sqrt{4 - x^2}}\right) = 2 \sqrt{4 - x^2} - \frac{2x^2}{\sqrt{4 - x^2}}$$ 4. **Set derivative to zero to find critical points:** $$2 \sqrt{4 - x^2} - \frac{2x^2}{\sqrt{4 - x^2}} = 0$$ Multiply both sides by $$\sqrt{4 - x^2}$$: $$2(4 - x^2) - 2x^2 = 0$$ Simplify: $$8 - 2x^2 - 2x^2 = 0 \Rightarrow 8 - 4x^2 = 0$$ $$4x^2 = 8 \Rightarrow x^2 = 2 \Rightarrow x = \sqrt{2}$$ (only positive root in domain). 5. **Evaluate area at critical point and endpoints:** - At $$x=0$$: $$A(0) = 0$$ - At $$x=2$$: $$A(2) = 2 \times 2 \times \sqrt{4 - 4} = 4 \times 0 = 0$$ - At $$x=\sqrt{2}$$: $$A(\sqrt{2}) = 2 \times \sqrt{2} \times \sqrt{4 - 2} = 2\sqrt{2} \times \sqrt{2} = 2\sqrt{2} \times \sqrt{2} = 2 \times 2 = 4$$ 6. **Conclusion:** The maximum area is $$4$$. The rectangle's dimensions at maximum area are: - Width: $$2x = 2\sqrt{2}$$ - Height: $$\sqrt{4 - x^2} = \sqrt{2}$$ --- 7. **Additional definitions:** Given functions: - Revenue: $$r(x)$$ - Cost: $$c(x)$$ - Profit: $$p(x) = r(x) - c(x)$$ Marginal values are derivatives: - Marginal revenue: $$\frac{dr}{dx}$$ - Marginal cost: $$\frac{dc}{dx}$$ - Marginal profit: $$\frac{dp}{dx} = \frac{dr}{dx} - \frac{dc}{dx}$$ These represent the rates of change of revenue, cost, and profit with respect to the number of items produced and sold.