Maximize Fencing for 4 Equal Pastures - 125,000 Linear Feet

[opus]

Homework Statement

A rancher has 125,000 linear feet of fencing and wants to enclose a rectangular field and then divide it into four equal pastures with three internal fences parallel to one of the rectangular sides. What are the dimensions for each of the four equal pastures that will maximize the area of each pasture?

Homework Equations

The Attempt at a Solution

Please see attached work. According to the back of the book, the dimension of 12,500 (green checkmark) is correct. But when I plug this value into the original equation to find the other dimension, I am getting an incorrect solution. Any ideas?

 

[magoo]

Under your step 1a, you had A = wl

Under step, 1b, you have 4A = wl. Obviously, it should be 4A = 4wl

OK?

 

[opus]

By multiplying a single side of the equation by 4, I understand that I changed the actual value of the equation. However, my reasoning is that I'm looking for 4 separate areas, each having $(w)(l)$. So why the 4 on the RHS as well? I feel like that makes the dimensions 4 times larger than what they should be.

 

[magoo]

A represents the area of one rectangle. You have 4.

 

[opus]

Yes which is why I have the 4 on the left. But wouldn't having the 4 on the right throw the dimensions of $w$ and $l$ off?

 

[ehild]

The question was:

What are the dimensions for each of the four equal pastures that will maximize the area of each pasture?

But you gave the length of the big rectangle.

 

[opus]

Ohh I see. Thank you guys. Then the second part should be 7812.5 ft.