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# A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?

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The sides perpendicular to the road will be $x$, and the one parallel to the road will be $y$

\begin{aligned} 2x+y&=2000\\ y&=2000-2x\\ \\ A&=x\cdot y\\ &=x(2000-2x)\\ &=-2x^2+2000x\\ \text{Find the vertex: }\\ x_{max}&=\dfrac{-b}{2a}\\ &=\dfrac{-2000}{-4}\\ &=500\\ \Rightarrow y&=2000-2x\\ &=1000\\ \\ \Rightarrow A&=x\cdot y\\ &=500\cdot1000=500000\end{aligned}

The maximum area is 500000 meters squared.

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