Question

A farmer owns 1000 m of fence and wants to enclose the largest possible rectangular area. the region to be fenced has a straight canal on one side, and thus needs to be fenced on only three sides. what is the largest area she can enclose and what are its dimensions?

Answer 1

These questions are hard until you "translate" them into maths. 
So this farmer wants to maximize the terratory enclosed by the fence. 
The area of a rectangle is a*b where we know that a+2b=1000. 
a+2b=1000
thus a=1000-2b 
a*b= b*(1000-2b) and you need to find the maximum for this
=-b^2 + 1000b 

Now you can do 2 things. Easy: use differentiation (if you know it) 
Hard: Use the method of completing a square