Suppose that a rancher has '1 000 feet of fencing available to make a rectangular corral. A barn will form one side of the corral so no fencing will be needed there. What dimensions will give the maximum area?

Before to get started , we are going to consider x= width of the rectangle and y=large of the rectangle.
Since a barn will form one side of the corral , there are only 3 sides required.Therefore the perimeter of rectangular corral will be:

Peremiter of the rectangle= 2x+2y

Perimeter= 2x+y=1000 (equation 1)

Remember that the perimterer of any polygon is the sum of its lenghts of all sides So 1000 ft of fencing available has to be the the sum of the lengths of the rectangular corral.

3. There are two unknown variables in the equation 1. Any of them can be solved . In this case we will choose y

y=1000 - 2x (equation 2)

Now In order to get the maximum area of the corral, we are going to define Area.

A(x,y)=x*y

4. Substitute y in the area equation.

A(x)=x*(1000-2x)

Then apply distributive property, therefore:

4. A=1000x-2x² (equation 3)

Since we want to get the maximum area, the next step is to find a local maximum of the function. This is possible through the first derivative =0 (first derivative test)

5. A'(x)=1000-4x=0

Then , solve for x

4x=1000

x=1000/4

x=250

6. Replace value of x in equation 2.

y=1000- 2(250)

y=500

In conclusion, the dimensions that will give the maximum area of the corral are: 250 ft and 500 ft