Thread: Optimization help

Hi everyone, I am currently taking Calc 1 and I am studying for a test we have tomorrow. I am doing an optimization problem and wanted to see if I am doing it correctly. Here it is:

A farmer has a plot of land that she wishes to turn into three pens. She has 2000 meters of fencing.
She wants to form three congruent rectangular pens by placing fencing.
What is the area of the largest field that she can fence?

I drew out a ling rectangle and divided it into 3 by placing two vertical lines inside the recatngle.

And this is how I think I solved it :

let the length of the whole field be y yds
let each divider line be x yds

so 2y + 4x = 2000
y = 1000 - 2x

area = xy
= x(1000-2x)
= 1000x - 2x^2


d(area)/dx = 1000 - 4x = 0 for a max area
4x=1000
x = 250
then y = 1000 - 2(250) = 500

The whole length should be 500 yards , and each smaller pen should be 250 yrads wide.
max area = 250(500) = 125000 yards^2


Since I have no other way of checking if I am doing this right any help/feddback on if this is correct or not would be greatly appreciated!

That is correct

Thanks for the verification!