A Developer Wants To Enclose A Rectangular Grassy Lot That Borders A City Street For Parking. If The Developer Has 344 Feet Of Fencing And Does Not Fence The Side Along The Street, What Is The Largest Area That Can Be Enclosed?


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Oddman answered
Let w represent the width of the fenced lot. That will be the distance between the street and the fence that is parallel to the street. The area will be given by
  area = w(344 - 2w) = -2w^2 + 344w
The extreme value of aw^+bw is obtained when w is -b/(2a). For your numbers, that value is
  w = -(344)/(2*(-2)) = 344/4 = 86
The area of the enclosed lot will be 86*172 ft^2 = 14,792 ft^2
For future reference in working such problems, note that the width is 1/4 of the perimeter fence length. Maximum area of a fully enclosed lot is obtained when the lot is square (or circular). Maximum area of a lot enclosed on 3 sides is obtained when the long side is parallel to the "open" side and is twice the length of the short side.

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