Find the length of the fence

Question

A rancher wants to fence in an area of 2500000 square feet in a rectangular field and then divide it in half with a fence down the middle, parallel to one side, as shown below.

What is the shortest length of fence that the rancher can use?

Answer 1

As Mark suggests:

You have a formula for area: LW, i.e. you can express L in terms of W

You have a formula for perimeter (P) which can be made into a function of the single variable W.

Differentiate P with respect to W....and there you have the answer.

Answer 2

Let's first give this rectangular fence some variables to represent its sides:

Let x represent the height. Let 2y represent the base.

Since there is a line of fencing that separates the fence into two pens, we essentially have two adjacent rectangles whose height is x and whose base is y.

What we know:

A = 2.5M (million)

A = bh (area equals base times height)

So, A = 2xy

P = sum of sides (for perimeter)

So, P = 3x + 4y

Let's rearrange our equations:

2.5M = 2xy

2.5M/(2y) = x

And, given P = 3x + 4y, we can substitute:

P = 3(2.5M/(2y)) + 4y = 3(2.5M)(2y)^(-1) + 4y

In any optimization problem, we must set the variable-dependent equation's derivative equal to zero and solve. (By variable-dependent equation, I just mean the equation we have less information about. For instance, we already know the value of A, but we dont know the value of P, so we will take the derivative of P and set it equal to zero.)

P' = -3(2)(2.5M)(2y)^(-2) + 4

Quick Review: the power rule implies that d/dx (x^n) = n(x)^(n-1); and don't forget to multiply by the chain!

Plug in P' into your calculator and find where the line crosses the x-axis.

Now we have y=968.25

Plug this value in to find x:

A = 2xy

2.5M = 2xy

2.5M = 2x(968.25)

2.5M/(2×968.25) = x

x = 1290.99

And finally, we can solve for the perimeter, or the amount of fencing:

P = 3x + 4y

P = 3(1290.99) + 4(968.25)

P = 7745.97

Therefore, the shortest length of fence the rancher can use, while still maintaining an area of 2,500,000 square feet, is 7,745.97 feet.