The greatest area that can be enclosed by a given amount of material will always be a square...
M=2L+2W so we can say:
2L=M-2W
L=(M-2W)/2
area is:
A=LW and using L from above:
A=W(M-2W)/2
A=(MW-2W^2)/2
dA/dW=(M-4W)/2
d2A/dW2=-4
Since acceleration, d2A/dW2, is a negative constant, when velocity, dA/dW=0, it is at an absolute maximum for A(W).
dA/dW=0 only when M-4W=0, 4W=M, W=M/4
from earlier we found L=(M-2W)/2 and using W from above we get:
L=(M-M/2)/2
L=(2M-M)/4
L=M/4
So L=W=M/4, thus it is a square.
That's the proof, anyway...
188=2x+2y
94=x+y
y=94-x
A=xy and using y from above:
A=94x-x^2
dA/dx=94-2x, d2A/dx2=-2, as explained earlier, since acceleration is a negative constant, when dA/dx=0, it is at an absolute maximum for A(x).
dA/dx=0 only when 94-2x=0, 2x=94, x=47
and from earlier, y=94-x, so y=94-47
y=47
So the dimensions that produce the greatest area are:
47ft by 47ft
And thus the greatest area is:
47^2=2209 ft^2
