Optimizing Fence Cost for Rectangular Field

[Calculus!]

1. The problem statement

A farmer wants to fence an area of 37.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence? (Give the dimensions in increasing order.)

The Attempt at a Solution

I was thinking to take 37.5 and divide it by two to get 18.75. So then one equation would be 18.75 = xy/2, right? I'm just not too sure how to get the equations.

 

[HallsofIvy]

Don't just do random things with numbers. The problem does NOT say that the internal fence divides the rectangle into equal parts which is what you seem to be thinking.

Draw a picture. Draw a rectangle with a line inside it parallel to one of the sides. Call the length x and the width y. What is the area? What is the length of fencing required?

 

[Calculus!]

I don't know. I'm still confused on what the picture will look like. I don't seem to understand.

 

[Calculus!]

but it does say "A farmer wants to fence an area of 37.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle."

 

[HallsofIvy]

Irrelevant. Putting the fence inside the original rectangle, parallel to one side, will divide it into two smaller rectangles, having the same total area and requiring the same fence length no matter where you put that internal fence.

Here's what the picture will look like: Draw a rectangle. draw a line inside the rectangle parallel to one side. Calll the length of the side perpendicular to the internal fence "x". Call the length of the side parallel to the internal fence "y". What is the total area in terms of x and y? Set that equal to 37.5 million square feet. What is the total length of fencing in terms of x and y? Minimize that.