Solve a Calculus I Fencing Problem: Minimize Cost

[rafael_josem]

Hi, I have the following problem of Calculus I class... I don't understand it and I don't know how to resolve it, can anybody help me??

A farmer wants to fence an area of 1.5x10^6 ft^2 in a rectangular field and then divide it to the half with a parallel fence at one side of the rectangle. How can he minimize the cost of the fence?

thanks...

 

[HallsofIvy]

Assuming that each "foot" of fencing costs the same, saying he wants to "minimize the cost of the fence" is the same as saying he wants to minimize the length of the fence.

Let L and W be the length and width of the rectangular field. Since the area of a rectangle is "length times width", we have LW= 1.5x10^6. If he had just a fence around the perimeter, it would extend along the length, the width, the length again, and then the width again: the length of the fence would be the perimeter of the rectangle: P= 2L+ 2W. However, we are told that the farmer is dividing the field in two by adding another fence parallel to one side of the triangle (NOT "at" one side- that is misleading). Taking it to be parallel to the width (it really doesn't matter which we call "length" and which "width") there is an additional fence length L. The total length of fencing is F= 3L+ 2W. This problem asks you to minimize F= 3L+ 2W with the condition that LW= 1.5x10^6.