+72 A farmer wishes to enclose a rectangular field that is already bounded on one side by a river (this side needs no fence). What are the dimensions of the field of maximum area that can be enclosed with 300 meters of fence)?
+33661 Let length = L, width = W and area = A
L + 2W = 300
A = L*W
A = (300 - 2W)*W or A = 300W - 2W^2
Using calculus:
dA/dW = 300 - 4W This is zero when W = 75, hence L = 300 - 2*75 or L = 150
Without calculus:
Completing the square:
A = -2(W^2 - 150W) → -2( (W - 75)^2 -75^2 ) This is a clearly a minimum when W = 75, hence L = 150.
.
+33661 Let length = L, width = W and area = A
L + 2W = 300
A = L*W
A = (300 - 2W)*W or A = 300W - 2W^2
Using calculus:
dA/dW = 300 - 4W This is zero when W = 75, hence L = 300 - 2*75 or L = 150
Without calculus:
Completing the square:
A = -2(W^2 - 150W) → -2( (W - 75)^2 -75^2 ) This is a clearly a minimum when W = 75, hence L = 150.
.
