Calculate the maximum area (maximum value) TX farmer has 100 metres of fencing to use to make a rectangular enclosure for sheep as shown.
He will use existing walls for two sides of the enclosure and leave an opening of 2 metres for a gate.
a)  Show that the area of the enclosure is given by: $A = 102x – x^2.$ 
b)  Find the value of x that will give the maximum possible area.
c)  Calculate the maximum possible area.
How do I assign the two variables for area ? Can anyone assist me in solving this problem?

From your picture, one side of the rectangle is x.
Since you have 100 metres, this means the other side has length (100 - x) + 2 = 102 - x.
So, the Area A = $(102 - x) * x = 102x - x^2$  
The maximum area occurs where $\frac{dA}{dx} = 102 - 2x = 0$
or where $x = 51$   
So, the max area A = $102(51) - 51^2$
