A farmer owns 1000 meters of fence, and wants to enclose the largest possible rectangular area. The region to be fenced has a straight canal on one side, and thus needs to be fenced on only three sides. What is the largest area she can enclose?

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A farmer owns 1000 meters of fence, and wants to enclose the largest possible rectangular area. The region to be fenced has a straight canal on one side, and thus needs to be fenced on only three sides. What is the largest area she can enclose?

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Answer 1 · 2015-06-20T02:45:10

I found: $A = 250 \times 500 = 125000 m^{2}$ $A=250xx500=125000m^2$

Explanation:

Considering the field as:
enter image source here
I know that the perimiter (only on 3 sides) to be fenced is equal to the meters of fence at disposal of the farmer:
$2 h + b = 1000 m$ $2h+b=1000m$ (1)
The area will be $A = b \times h$ $A=bxxh$ (2)

From (1) $b = 1000 - 2 h$ $b=1000-2h$ in (2)

$A = (1000 - 2 h) \times h = 1000 h - 2 h^{2}$ $A=(1000-2h)xxh=1000h-2h^2$

Derive $A$ $A$ with $h$ $h$ :

$A'=1000-4h$
equal it to zero to maximize it:

$1000-4h=0$

$h=1000/4=color(red)(250m)$
use this back in (1) you find $b=color(red)(500m)$ :
Use these dimensions in (2): $A=250xx500=color(blue)(125000m^2)$

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A farmer owns 1000 meters of fence, and wants to enclose the largest possible rectangular area. The region to be fenced has a straight canal on one side, and thus needs to be fenced on only three sides. What is the largest area she can enclose?

1 Answer

Explanation:

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