A rancher wants to fence in an area of 500000 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?

Answer 1

The last answer was for basic algebra. If you mean to solve this for a calculus class, this is how to do it. You need to optimize a function. We first have to start out with definitions... Let L represent the length of the field.

Let W represent the width of the field.

Let F represent the length of fence needed.

L*W=500000

2L+3W=F Both length sides+both width sides+ fence across the width.

Let's look at the first equation again:

L*W=500000

W=(500000/L)

Now we can substitute (500000/L) for W in the second equation.

2L+3(500000/L)=F

2L+1500000/L=F

Now we have a function. We can input any length, and we'll know length needed. But how can we find the length needed so that F is as small as it can be? If you know derivatives, there's an exact way. The next easiest way to do it is to graph.

Manner 1: Derivative

Set the derivative to zero...when the length stops going down and starts coming back up, it'll be at it's lowest point.

2L+1500000^-1=F

F'=2-1500000^-2

0=2-1500000^-2

1500000/(L^2)=2

1500000=2L^2

1500000=L^2

L=866.025

Now, we know that L=500000/W

so 866.025=500000/W

W=500000/866.025

W=577.35027 feet.

Now,

2L+3W=2(866.025)+3(577.35027)

Minimum Fencing needed: 3464.1016 feet.

(To point out why a square doesn't work... Since you have three sides, it's more optimum to make this a shorter length, and make the two sided length slightly longer.)

Method 2: Graphing. Graph y=2x+1500000/x (y=fence size, x=length of side)

to to the "bottom" or the graph. Zoom in really far. As we zoom in closer and closer, we find that x=866.025 and y=3464.1016.

Thus, the minimum amount of fencing needed is 3464.1016 feet.