Calculus: Optimization word problem. Help needed!?

Question

You are to enclose 3 sides of a rectangular area of 512 square feet with fencing. If fencing costs are $2.00 per foot, what are the dimensions of the rectangle that will minimize the total cost of fencing and still enclose the required area? What is that minimum cost?

Please show all work, and if possible, describe each step. The more description, the better. Thanks.

Answer 1

let the length and width be 'x' and 'y' reply.
given xy=512
so x=512/y
the perimeter=2x+y
=2x+512/x
for optimising
dP/dx=0
2-512/x^2=0
multiplying by x^2
2x^2-512=0
2x^2=512
dividing by 2
x^2=256
x=rt256=16
y=512/x=512/16=32
so the dimensions for min perimeter
=16'*32'
perimeter=2x+y=64'
cost=64*2=$128

Answer 2

A = xy = 512
L = x + 2y
C = $2*L
x = 512/y
L = 512/y + 2y
To find minimum L (and thus minimum C):
dL/dy = -512/y² + 2 = 0
-512/y² = - 2
512/y² = 2
512 = 2y²
y² = 521/2 = 256
y = 16 ft.
x = 512/16 = 32 ft.
L = 32+2*16 = 64 ft.
C = 2*64 = $128