Question

A rancher wants to fence in an area of 3000000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use?

Answer 1

welll you have the lenght of the fence that will bi equal to 2 the lenght plus 3 times de width (2L+3W) and you know that WxL=3000000 so if you take F(L)=2L+ 3(3000000/L) all you need is to take de derivative and make F´(L)=0 then you just solve for W

Answer 2

i see where i misunderstood, you want the shortest total length not just the fence down the middle. ;)

Answer 3

This is a simple optimization calculus problem. It helps to draw a picture (not sure how good this will come out)
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That's what we're trying to build, and we want to minimize the fence used. We'll call the length a and the width b.

The length of fence we will call P.

P = 2a + 3b

We also know the goal area:
A = a*b
a*b = 3,000,000

solve for b in this equation:
b = 3,000,000/a

plug into the other equation

P = 2a + 3(3,000,000)/a

Now we have the find the derivative of this and set it equal to zero.

P' = 2 - 9,000,000/a^2
0 = 2 - 9,000,000/a^2
2a^2 = 9,000,000
a = +/-sqrt(9,000,000/2)

a, is the length of the area being contained, so it can't be negative. So

a = sqrt(9,000,000/2)

plug this value into the equation we had for P in terms of a:
P = 2a + 3(3,000,000)/a

and we get the minimum length of fence = 2(sqrt(9,000,000/2)) + 9,000,000/(sqrt(9,000,000/2))

Simplify as needed.