+130516 Let x be the length of the side of any of the pens.....the width of one side = w
The total perimeter is given by 4x + 6w and this equals 360
So.....
4x + 6w = 360 solving for w, we have
6w = [360 - 4x]
w = [360 - 4x] / 6 = 60 - (2/3)x
So....since the area of each pen is the same,we want to maximize any one of them
And the area, A, of one pen = l * w = x * [ 60 - (2/3)x].......
So we have
A = x [ 60 - (2/3)x]
A = -(2/3)x^2 + 60x
The x value that maximizes the area is given by x = - b/ 2a where b = 60 and a = -2/3
So
x = -60 / [2( -2/3)] = -60/(-4/3) = 60/(4/3) = 60 *3/4 = 45 ft this is the length of one of the pens
The width is given by :
[60 - (2/3)(45) ] = [60 - 30 ] = 30 ft
So....the maximum area of each pen is when the width = 30 ft and the length = 45 ft
And the max area of one of the pens = l * w = 45 (30) = 1350 sq ft
