Question

You have 500-foot roll of fencing and a large field. You want to construct a rectangular playground area. What are the dimensions of the largest such yard? What is the largest area?

Answer 1

A square of `125 feet` sides
Area = `125^2 = 15625 feet^2`

Explanation:

Upon investigation you will discover that the greatest area for any particular circumference is that of a square.

So the length of one side is `500/4 = 125 feet`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Lets the lengths of the sides be `b and c -> 2b+2c=500`

Thus `b=(500-2c)/2=250-c`

Area`->a=bxxc color(white)("dddd")-> color(white)("dddd")(250-c)xxc`

`color(white)("dddddddddd.")" Area "color(white)("d")->color(white)("dddd")a=250c-c^2" "....Eqn(1)`

As the `c^2` term is negative then the graph is of form `nn` thus it has a maximum and this is the vertex.

However, we need to write `Eqn(1)` in the form of:

`y=0=+c^2-250c+a`

using part of the proces of completing the square we have:

`c_("vertex")= (-1/2)xx(-250) = +125`

Substitute this back into `Eqn(1)`

`a=250(125)-(125^2) = 15625 feet^2`