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?

↳Redirected from "How to prove this ? #2|k^2# then #2|k# for some #k\inZZ#"
1 Answer
Gió
Jun 20, 2015

I found: #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:
#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)#

Related questions
See all questions in Solving Optimization Problems
Impact of this question
23791 views around the world
You can reuse this answer
Creative Commons License