Question

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?

Answer 1

I found: `A=250xx500=125000m^2`

Explanation:

Considering the field as:

I know that the perimiter (only on 3 sides) to be fenced is equal to the meters of fence at disposal of the farmer:
`2h+b=1000m` (1)
The area will be `A=bxxh` (2)

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

`A=(1000-2h)xxh=1000h-2h^2`

Derive `A` with `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)`