   SEARCH HOME Math Central Quandaries & Queries  Question from Kate, a student: A farmer has a problem with rabbits and skunks in his rectangular carrot patch that is 21m^2 in area. Determine the dimensions that will require the least amount of fencing if a barn can be used to protect one side of the garden. Hi Kate.

You know the area and you know you need to fence three sides.

Let P = the length of fence parallel to the side of the barn and let F be the total length of fencing that you need.

That means the other side of the rectangle is (F-P)/2 units long.

The area of a rectangle is the length times the width, so this is P times (F-P)/2. You know the area is 21 square meters.

So:

P (F-P) / 2 = 21.

When you multiply this out, you have a quadratic relationship. You are trying to find the minimum F.

Think about parabolas which are quadratic relationships. They have a minimum point at the vertex. Can you write the equation in vertex-graphing form and find the vertex?

That will give you the correct values of P and F, which you can use to determine the other dimension of the carrot patch using (F-P)/2.

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.