Area and Perimeter

Area and perimeter share a quadratic relationship. Consider a rectangular field of length L and width W which you wish to fence in using 100 meters of fencing. Using the fact that the 100 meters is the perimeter of the rectangle, solve for W in terms of L and substitute this into the formula for the area of the rectangle (giving you a quadratic in the variable L.)

Graph this quadratic on graph paper.

For what range of lengths is the area positive?

For what lengths is there no enclosed area? (In other words, where are the x-intercepts?)

For what length is the area greatest? (In other words, where is the vertex of the parabola?)

What are the dimensions of the field with greatest area? What type of rectangle is this?

Given an arbitrary perimeter P find a formula (in terms of P) for the maximum area and the dimensions of the field that can be enclosed?

On another piece of property there is already a fence on one side. If you have 100 meters of fencing, what is the maximum area that can be enclosed? What dimensions give you the maximum area?