You have 96 feet of fencing to enclose a rectangular region, what is the maximum area of the region?

max area is if it is a square. Then one side = 96/4 = 24 ft

and area is 24² = 576 sq ft

Half the length of the fencing is 48 feet. 

Hence Area(A) = x(48 - x) 

Where 'x' is the unknown value of one side and and 48 - x is the length of the other side. 

Hence 

Differentiate and equate to zero . 

DA / dx = 48x - x^2  = 0 

48 - 2x = 0 ;  x = 24 is the length of the unknown side 'x' & 48 - x = 24 the length of the other unknown side 

Since the two sides are each 24 ft , then the rectangle is a square

The Area (A) = 24ft x 24ft = 576 sq.ft. which is the maximum area of the region. 

NB If you change the dimensions to say 20 x 28 = 560 sq.ft. ,which is less area.  So a square gives the maximum area. 

96/4=24

24*24=576 sq ft.

 If you have 96 feet of fencing to enclose a rectangular region, 

 what is the maximum area of the region?

 The maximum area of the region is (24^2) or 576 feet^2.

I hope that no one will help you cheat...

Let x & w be the length & width resp. Thus x+x+w+w = 96.

Thus w= 48 - x. Convince yourself of this. 

The area is thus A =  x(48-x). 

This is a downward parabola. Therefore it has a max which is its vertex. And you should know how to find the coordinates of the vertex of a parabola. Or use calculus to find its derivative and set to zero.

Done!

Show your steps here if need be and we can go through them with you.

Hopefully no one will spoil you the answer. That would be very irresponsible of them. And don't forget to vote me best answer for being the first to correctly walk you through without spoiling the answer. That way it gives you a chance to work at it and to get good at it!