How Can a Farmer Minimize the Cost of Fencing a Rectangular Field?

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  • Thread starter karush
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in summary, the farmer wants to fence an area of 1500 ft^2 in a rectangular field with a fence that is 1000 ft long and costs $1500$.
  • #1
karush
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this is an optimization problem

A farmer wants to fence an area of $1.5$ million square feet
$(1.5\text{ x }10^6 \text{ ft}^2)$
in an a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?

well, I understand the problem to mean that the fence goes around the perimeter then another fence goes down the middle dividing the area in half. So using the length $l$ as $2$ sides and another fence $l$ length going down the middle then.

total length of fence $\displaystyle f(l) = 3l+\frac{2\cdot 1.5\text{ x }10^6}{l}$

persuming this is correct then $\frac{d}{dl}f(l)=0$ would be $l$ for the min length of fence for that area
 
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  • #2
I would let $0<x$ represent the 3 lengths of fence and $0<y$ represent the two lengths perpendicular to the first 3. And so the total length $L$ of fence is:

\(\displaystyle L(x,y)=3x+2y\)

This is our objective function, i.e., that which we wish to optimize.

Now, we are constrained by the fixed area $0<A$ to be enclosed, and since the area is rectangular, we may write:

\(\displaystyle A=xy\)

Solving the constraint for $y$, we obtain:

\(\displaystyle y=\frac{A}{x}\)

Hence, we may write the length function in terms of one variable:

\(\displaystyle L(x)=3x+\frac{2A}{x}\)

This is equivalent to what you have. Now, when optimizing, you do want to differentiate with respect to the independent variable and equate the result to zero, and I would also use one of the derivative tests to demonstrate that the critical value found is indeed at a minimum.

Note: I use $A$ simply so that the large number can be plugged in at the end, after the calculations are done, instead of writing it numerous times. In your third semester of calculus, you will be taught a method for problems like this that is computationally much simpler, called Lagrange multipliers. :D
 
  • #3
I changed this to avoid 2 variables

$L(x) = 3x-3\cdot 10^6\cdot x^{-1}$

so $\displaystyle\frac{d}{dx}L(x)$ is $\displaystyle 3-\frac{3\cdot 10^6}{x^2}$

then $3x^2-3\cdot 10^6=0$ so $x=1000\text { ft}$

and $\displaystyle\frac{1.5\cdot 10^6}{1000}=1500\text{ ft}$
 
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FAQ: How Can a Farmer Minimize the Cost of Fencing a Rectangular Field?

1. What does "minimize the cost of the fence" mean?

"Minimize the cost of the fence" means finding the most cost-effective solution for building a fence, taking into consideration factors such as materials, labor, and other associated costs.

2. What factors should be considered when trying to minimize the cost of the fence?

Factors that should be considered include the materials used for the fence, the length and height of the fence, the terrain and location of the fence, the labor costs, and any additional costs such as permits or equipment rental.

3. How can I determine the most cost-effective materials for my fence?

To determine the most cost-effective materials, you can research the prices of different materials and compare them. You should also consider the durability and maintenance costs of each material to determine the long-term cost-effectiveness.

4. Is it better to hire a professional or build the fence myself to minimize costs?

This depends on your own skills and experience. If you are confident in your abilities and have the necessary tools and resources, building the fence yourself may be more cost-effective. However, hiring a professional may save you time and ensure a higher quality fence.

5. Are there any other ways to minimize the cost of the fence besides materials and labor?

Yes, there are other ways to minimize the cost of the fence. For example, you can adjust the design or layout of the fence to reduce the amount of materials needed. You can also consider purchasing materials in bulk and negotiating prices with suppliers.

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