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

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
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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