Let x and y be the length and width of the rectangular corral.

Since the two rectangular corrals are adjacent to each other, let one side of each corral (x) share a common fence.So, their total perimeter is:

P=3x+4y

Plug-in the given perimeter of the fence.

200=3x+4y

Then, isolate either of the variable. Let the y be isolated.

y = (200-3x)/4

Next, set-up the equation for total area of the two rectangular corrals.

A=2(xy)

plug y = (200-3x)/4 into the last equation:

A=2x(200-3x)/4

A=x(200-3x)/2

A = (200x-3x^2)/2

A = -(3/2)x^2 + 100x

we need to find the maximum of the parabolic function A = -(3/2)x^2 + 100x

since the quotient in front of x^2 is -3/2 < 0 the function has maximum in its vertex:

x max = - b/2a, where a = -3/2, b = 100

x max = -100/(2*(-3/2))

x max = 100/3

x max = 33 1/3 ft

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y = (200-3(100/3))/4

y = 25 ft

to produce the largest total area the dimensions of each corral must be: 33 1/3 ft and 25 ft.