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?

1 Answer
Apr 10, 2018

Answer:

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#