Find the dimensions of the rectangular field to minimize the cost of the fence?

Question

A farmer wants to fence an area of 2 million square feet in a rectangular field and then divide it into three equal parts with two fences parallel to one of the sides of the rectangle (see the below figure). Find the dimensions of the rectangular field to minimize the cost of the fence?

Answer 1

Let x and y represent the width and length of the rectangle surrounding the field.

xy = 2(10⁶) ft² (area of a rectangle is length times width).(x and y both positive)

The cost of each foot of fencing is not given, but the cost can be minimized by determining the least amount of fencing that can be used to enclose two million square feet.

Let x be the side that is replicated twice inside the rectangle to create the three sections.

Then the amount of fencing that is required in terms of x and y is:

F(x,y) = 2y + 4x

But y = 2(10⁶)/x = 2(10⁶)x^-1

So,

F(x) = 2[(2)(10⁶)]x^-1 + 4x

So F'(x) = -4(10⁶)x^-2 + 4

Set that equal to zero to determine critical numbers:

-4(10⁶)/x² + 4 = 0

-4(10⁶) + 4x² = 0

4x²= 4(10⁶)

x²=10⁶

x=±1000 (rejecting the negative value, x = 1000)

y = 2(10⁶)/10³ = 2(10³) = 2000

xy = (1000)(2000) = 2000000 ft²

Amount of fencing required:

F(1000,2000) = 2(2000) + 4(1000) = 8000 feet of fencing is the minimum required to fence in a rectangular field of two million square feet.

desmos.com/calculator/dvllp4deol