Max Area River D8Ddec

1. **State the problem:** A farmer has $4000$ meters of fencing to enclose a rectangular plot next to a river. The side along the river is not fenced. We want to find the dimensions that maximize the enclosed area. 2. **Define variables:** Let $x$ be the length of each vertical side (perpendicular to the river). The side along the river is not fenced, so the fencing is used for the two vertical sides and the bottom side. 3. **Write the fencing constraint:** The total fencing used is for two vertical sides and one bottom side: $$2x + (4000 - 2x) = 4000$$ But the bottom side length is $4000 - 2x$ as given, so the fencing length is: $$2x + (4000 - 2x) = 4000$$ This confirms the fencing length is $4000$ meters. 4. **Express the area $A$ of the rectangle:** The area is height times base: $$A = x \times (4000 - 2x)$$ 5. **Simplify the area function:** $$A = 4000x - 2x^2$$ 6. **Find the maximum area:** Since $A$ is a quadratic function opening downward (coefficient of $x^2$ is negative), the maximum occurs at the vertex. The vertex $x$-coordinate is given by: $$x = -\frac{b}{2a} = -\frac{4000}{2 \times (-2)} = \frac{4000}{4} = 1000$$ 7. **Calculate the maximum area:** Substitute $x=1000$ into the area formula: $$A = 4000(1000) - 2(1000)^2 = 4,000,000 - 2,000,000 = 2,000,000$$ 8. **Interpretation:** The largest area that can be enclosed is $2,000,000$ square meters when each vertical side is $1000$ meters and the bottom side is: $$4000 - 2(1000) = 2000$$ meters. **Final answer:** $$\boxed{2,000,000 \text{ square meters}}$$