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

In summary, the conversation is about a rancher who has 125,000 linear feet of fencing and wants to enclose a rectangular field and divide it into four equal pastures. The goal is to find the dimensions for each pasture that will maximize their area. The solution involves finding the area of one rectangle, represented by A, and then multiplying it by 4 to get the total area of the four pastures. The correct dimension for each pasture is 12,500 feet, which results in an area of 7812.5 feet for each pasture. There was some confusion about the steps in the solution, but it was eventually clarified that the dimensions given were for the big rectangle, and the dimensions for the four equal pastures
  • #1
opus
Gold Member
717
131

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?
 

Attachments

  • Screen Shot 2019-01-04 at 7.29.14 PM.png
    Screen Shot 2019-01-04 at 7.29.14 PM.png
    30.5 KB · Views: 283
Physics news on Phys.org
  • #2
Under your step 1a, you had A = wl

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

OK?
 
  • Like
Likes opus
  • #3
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.
 
  • #4
A represents the area of one rectangle. You have 4.
 
  • Like
Likes opus
  • #5
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?
 
  • #6
The question was:

opus said:
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.
 
  • Like
Likes opus
  • #7
Ohh I see. Thank you guys. Then the second part should be 7812.5 ft.
 

1. What is the purpose of maximizing fencing for 4 equal pastures with 125,000 linear feet?

The purpose of maximizing fencing for 4 equal pastures with 125,000 linear feet is to efficiently divide a large area of land into four equal sections for the purpose of grazing livestock or farming. This allows for better management and rotation of the pastures, leading to healthier and more productive land.

2. How do you determine the optimal length of fencing for each pasture?

The optimal length of fencing for each pasture can be determined by dividing the total length of fencing (125,000 linear feet) by 4, resulting in 31,250 linear feet for each pasture. This ensures that each pasture is equal in size and has an equal amount of fencing for efficient management.

3. What factors should be considered when maximizing fencing for 4 equal pastures?

When maximizing fencing for 4 equal pastures, factors such as the terrain, soil type, and location of water sources should be considered. These factors can affect the layout and placement of fencing, as well as the overall cost and effectiveness of the fencing.

4. What type of fencing is recommended for maximizing 4 equal pastures?

The type of fencing recommended for maximizing 4 equal pastures will depend on the intended use and budget. Options include traditional post and wire fencing, electric fencing, or high-tensile fencing. It is important to choose a fencing type that is durable, cost-effective, and suitable for the type of livestock or crops being managed.

5. How can maximizing fencing for 4 equal pastures benefit the environment?

Maximizing fencing for 4 equal pastures can benefit the environment in several ways. By rotating livestock and crops between pastures, the land can rest and recover, leading to healthier soil and increased biodiversity. Additionally, properly managed fencing can prevent overgrazing and soil erosion, leading to improved water quality and conservation of natural resources.

Similar threads

Minimum cost of an area of fencing using derivatives
    • Calculus and Beyond Homework Help
Replies
3
Views
3K
Gr 12 Calc: Optimization problem/min cost
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Maximize Area of Rectangle w/ 2400ft Fence
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Optimization problem involving an area and perimeter
    • Calculus and Beyond Homework Help
Replies
3
Views
3K
Maximize Area of Rectangle w/ x Feet Fencing
    • Calculus and Beyond Homework Help
Replies
3
Views
2K
Maximizing a rectangle in R3
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
Optimizing Fence Cost for Rectangular Field
    • Calculus and Beyond Homework Help
Replies
4
Views
5K
Maximizing Enclosed Area with 400ft Fencing
    • Calculus and Beyond Homework Help
Replies
6
Views
7K
Quadratic Functions Word Problem
    • Precalculus Mathematics Homework Help
Replies
7
Views
3K
I Optimization problem with multiple outputs: impossible?
    • General Math
Replies
6
Views
1K

Hot Threads

Recent Insights

Back
Top