If I have 40 feet of fencing and i need to build a four sided enclosure what are the dimensions which give me maximum area?

Question

sburchette · Answer

We can use x and y for the dimensions of the fence. The perimeter of the fence will always be 40, so
$$2x+2y=40$$
The area is simply
$$A=xy$$ Which is the area we wish to maximize.  We can solve for x or y in the first equations and then substitute that into the area equation so that it is in terms of one variable.

$$y=20-x$$ So substituting 20-x for y in the area equation we get
$$A=x(20-x)$$ Distributing x we get
$$A=-x^2+20x$$
To find the max of this function we can differentiate A and find the critical numbers.
$$A'=-2x+20$$
Critical numbers (and extrema) occur where the derivative is equal to 0.
$$0=-2x+20$$
$$x=10$$
So the area is maximized when x=10.  Using our perimeter equation we can determine that y=10
So the dimensions that give a maximum area are 10 and 10.

anonymous · Answer

great stuff - thanx
so its always a square that gives maximum area