Question

Mr peterson will buy 80 feet of fencing. The pieces of fence come in whole numbers of feet. His enclosure will be a rectangle and have 4 sides. If he wants to enclose the least area, what should be the length, width, and area?

Answer 1

The perimeter is:
 P = 2x + 2y = 80
 The area is:
 A = x * y
 The area as a function of a variable is:
 A (x) = x * (40-x)
 Rewriting we have:
 A (x) = 40x-x ^ 2
 We derive the function:
 A '(x) = 40-2x
 We equal zero and clear x:
 40-2x = 0
 2x = 40
 x = 40/2
 x = 20 feet
 The other dimension is:
 y = 40 - x
 y = 40 - 20
 y = 20 feet
 The area will be:
 A = x * y
 A = 20 * 20
 A = 400 feet ^ 2
 Answer:
 The dimensions are:
 x = 20 feet
 y = 20 feet
 The area is:
 A = 400 feet ^ 2