Suppose you wish to enclose a field for corn corn by a fence the is 2x feet long and y feet wide. The perimeter of the fence is 800 feet. What is x and y and what dimensions will maximize the area of the fence?
Perimeter: 2x + 2x + y + y = 800
4x + 2y = 800
Area = 2x(y)
Solve for y in the equation for perimeter and substitute in the equation for area
2y = 800 - 4x
y = 400 - 2x
Area = 2x(400 - 2x)
= 800x - 4x^2
To maximize the area, take the derivative of the area and set equal to 0 and solve for x.
Derivative of Area = 800 - 8x
Therefore, x = 100
y = 400 - 2(100)
The dimensions of the field which gives maximum area is 200 x 200 (which makes sense, a square will always maximize area)