Geometry problem I do not understand how I will have different dimentions and what the configuration is

Question

Many real-world problems involve the idea of optimization. For example, if I need to buy fencing to create a 250,000 square foot enclosure (almost 6 acres) for my horses but I want to spend the least amount of money possible, I would use optimization to determine the dimensions that would use the least amount fencing (perimeter) while giving me an enclosed area of 250,000 square feet. In this activity, you will choose an animal that you need to make a rectangular fenced enclosure for. You will research the square footage needed for the animal's enclosure. Using the required square footage for the enclosure, you will determine the dimensions that will require the least amount of fencing to fence in the perimeter of the rectangular enclosure (you must start with the required square footage, from which you will determine the dimensions that will require the least amount of fencing). You will explain your procedure for finding the dimensions and how the amount of fencing required changes as the dimensions change. 

-State what animal you chose and what the minimum required square footage (area) is for the fenced enclosure (do not include dimensions, only the total required square footage). Your answer must include a minimum of 1 complete sentence.

-Provide the width and length for 3 different possible configurations of your enclosure (different dimensions but same square footage). Draw a picture that represents a scaled version of the approximate shape of each configuration. For each of the three possible configurations show the calculation of the area using the dimensions you provided to ensure that it is providing the proper square footage AND calculate the perimeter for each, showing your work.

-Circle the configuration in part 2 that requires the least amount of fencing to fence the perimeter of the enclosure. Describe in at least two complete sentences what you notice about the shape of this enclosure and how it compares to the other two, less optimal, configurations. (1 point)

Answer 1

the largest area is always a square

you have 250,000 square feet

take the square root of this number

√250,000=500

you want a square 500' x 500'

4*500=2000 feet of fencing

here's the work but it involves derivatives and I don't know if you are familiar with derivatives

A=lw

P=2l+2w

250,000=lw

l=250,000/w

P=2(250,000/w) +2w

P=(500,000/w)+2w

P=500,000w^-1+2w

take the first derivative

P'=-500,000w^-2+2

set the derivative equal to zero

-500,000w^-2+2=0

500,000w^-2=2

500,000/w²=2

2w²=500,000

w²=250,000

take the square root of both sides

w=500 feet again

you have a square 500' by 500'

4*500=2000 feet