Question

suppose you have to use exactly 200m of fencing to make either one square enclosuee or two separate square enclosures of any sizes you wish.what plan gives you the least area?the greatest area?

Answer 1

Assume one square enclosure of side x.
Perimeter of the square with side x is: 4x
4x = 200 m
x = 50 m

Area = x^2 = ?

Answer 2

Assume two square enclosures, of sides y and z respectively.
Total perimeter of the two squares, one of side y and another of side z is: 4(y + z)
4(y + z) = 200
y + z = 50 --- (1)

Total Area A = p^2 + q^2 = p^2 + (50 - p)^2
A = p^2 + 2500 - 100p + p^2
A = 2p^2 - 100p + 2500
This is a vertical parabola that opens upward and it has a minimum value at the vertex.
What are the x and y coordinates of the vertex. The y-value will be the least area.
The area calculated in the previous reply will be the greatest area.