Show that the rectangle of a perimeter is always a square.

Question

Show that the rectangle of a minimum perimeter for a given area A is always a square.

Answer 1

A = LW ⇒ L = A/W

 

P = 2L + 2W = 2(A/W) + 2W

 

dP/dW = 0 = -2A / W^2 + 2 ⇒ A = W^2 = LW ⇒ W = L, hence L=W makes it a sqaure