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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?


Answered 2 years ago
Answered 2 years ago
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The sides perpendicular to the road will be xx, and the one parallel to the road will be yy

2x+y=2000y=20002xA=xy=x(20002x)=2x2+2000xFind the vertex: xmax=b2a=20004=500y=20002x=1000A=xy=5001000=500000\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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