Adog owner wants to use 200 ft of fencing to enclose the greatest possible area for his dog. he wants the fenced area to be rectangular. what dimensions should he use?

Perimeter (P) = 2 · Length(L) + 2 · Width (W) → P = 2L + 2W
Solve for either L or W (I am solving for L).
200 - 2W = 2L
(200 - 2W)/2 = L
100 - W = L

Area (A) = Length (L) · Width (W)
              = (100 - W) · W
              = 100W - W²

Find the derivative, set it equal to 0, and solve:
dA/dW = 100 - 2W
     0    = 100 - 2W
     W  = 50

refer to the equation above for L: 
100 - W = L
100 - 50 = L
        50  = L

Dimensions for the maximum Area are 50 ft x 50 ft