Bob has 50 feet of fencing to enclose a rectangular garden. if one side of the garden is x feet long, he wants the other side to be (25 – x) feet wide. what value of x will give the largest area, in square feet, for the garden?

The area will be:

A=x(25-x)

A=25x-x^2

For the area to be maximize the rate of change of area will be zero, or dA/dx=0

dA/dx=25-2x

25-2x=0

2x=25

x=12.5

So the dimensions will be 12.5 and (25-12.5)=12.5.  Thus the greatest area possible with 50 foot of fencing is a square with sides of 12.5 feet.

(A square always results in the greatest possible area for a rectangular plane for a given amount of material...so in general, all such problems will result with dimensions of a square with sides equal to the material divided by four.)