# 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?

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 = $102(51) - 51^2$