Optimization problem

[aliswan]

A farmer wants to fence an area of 1.5 million square feet in 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?

i know one of my equations is
A=xy=1500000

but other than that i have no idea how to solve it

I am also completing this problem on the Maple 13 software.

 

[tkhunny]

Length of fencing = 2x + 3y

Shall we presume that all fencing costs the same? We just minimize the fencing and forget about the cost.

 

[TchrWill]

A farmer wants to fence an area of 1.5 million square feet in 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?

i know one of my equations is
A=xy=1500000

but other than that i have no idea how to solve it

Considering all possible rectangles with a given area, the square has the smallest perimeter

Proof:
Consider a square of dimensions "x "by "x", the area of which is x^2.
Subtract "a" from one side making it (x - a).
Add "b" to the other side making it (x + b).
Since x^2 - ax + b(x - a) = A, "b" must be greater than "a" as "ax" must equal b(x - a).
Therefore, P = 2(x - a) + 2(x + b) = 2x - 2a + 2x + 2b = 4x - 2a + 2b.
With "b" greater than "a", 4x - 2a + 2b results in a greater perimeter P.
Consider the family of rectangles with area 36 sq.units.
The rectangle dimensions and their associated perimeters are:
x......1......2......3......4......6
y.....36....18....12......9......6
P....74....40....30.....26....24 showing that the square has the smallest perimeter.

What is the perimeter of a square field of 1,500,000 sq.ft. with a dividing fence down the middle?

What is the perimeter of two adjoining squares each of 750,000 sq.ft.

 

[srmichael]

A farmer wants to fence an area of 1.5 million square feet in 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?

i know one of my equations is
A=xy=1500000

but other than that i have no idea how to solve it

I am also completing this problem on the Maple 13 software.

This problem screams derivatives.

Let x=length of the rectangle and y=width of the rectangle.

\(\displaystyle xy=1,500,000\)

So if I divide the rectangle down the middle with another length of "y' you get the perimeter formula of:

\(\displaystyle P=2x+3y\). This is what you are trying optimitize and thus what you need to take the derivative of. Before you take the derivative of this, however, you may want to rewrite it all in terms of one variable. Enter the \(\displaystyle xy=1,500,000\) formula. Solving for "y" you get \(\displaystyle y=\frac{1,500,000}{x}\). You could have solved for "x" if you had wanted to as well.

Plug this into the perimeter formula and you get:

\(\displaystyle P=2x+3(\frac{1,500,000}{x})\).

Taking the derivative of this, set it equal to zero and see what you get.