A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 548 feet of fencing is used. How do you find the dimensions of the playground that maximize the total enclosed area?

1 Answer
Alan P.
Oct 23, 2015

137 xx 91 1/3 feet

Explanation:

Let
color(white)("XXX")L= length of entire playground
color(white)("XXX")W= width of entire playground.

Assume the dividing fence is parallel to the width.

Total fencing required
color(white)("XXX")2L + 3W = 548

color(white)("XXX")L= (548-3W)/2

Area of entire playground
color(white)("XXX")A = LxxW

color(white)("XXXX")=(548-3W)/2*W

color(white)("XXXX")= -3/2W^2+274W

Maximum area will occur at a point where the derivative of the area is equal to zero.

color(white)("XXX")(dA)/(dW) =-3W+274 = 0

color(white)("XXX")W=91 1/3

Substituting 91 1/3 for W in 2L+3W=548
gives
color(white)("XXX")L=137

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