SOLUTION: A farmer wants to fence in a field using the river as one side of the enclosed area. The farmer wants 5 pens and uses 2000 feet of fencing. What are the dimensions of the area if t

Let x be the size of the pens perpendicular to the river.


Then there are 6 sides of the pens perpendicular to the river 
and one long side parallel to the river.


The length of the parallel side to the river is 2000-6x.


The fenced area is the area of the rectangle with the sides x and (2000-6x)

    Area = A(x) = x*(2000-6x)


It is a quadratic function with x-intercepts at x= 0  and  x=  = 333.33.


It has the maximum at x at the mid-point between the x-intercepts.


So,  x =  = 166.67 ft is the optimal size of the pens perpendicular to the river.


The length of the fenced field along the river is 2000-6x = 2000 -  = 2000 - 1000 = 1000 ft.


This length of 1000 ft is divided by 5 pens - so each pen is 200 ft along the river.