# 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?

## 1 Answer

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 = $102(51) - 51^2$

• You're welcome. – user137481 Mar 17 '15 at 21:06
• thank you for your assistance and guiding me in the solution. Area=length x width Width = x Length of the enclosure parallel to the wall = (100-x) +2 =102 - x Area= x(102-x) Area, A = 102x-x^2 dA/dx=102- 2x At Stationary Point dA/dx=0 ⇒ 102 – 2x = 0 2x=102 x=51 the maximum area occurs when x = 51 and Area, A= (102x-x^2) = 102x51 – (51)^2 = 2600 – MsK05 Mar 17 '15 at 22:14