A rectangular bird sanctuary is being created with one side along a straight riverbank. The remaining three sides are to be enclosed with a protective fence. If there are 12 km of fence available, find the dimension of the rectangle to maximize the area of the sanctuary.

For a rectangle of length L and width W, the perimeter is:

P = 2*L + 2*W

And the area is:

A = L*W.

Now, we know that the sanctuary has one side along a straight riverbank, then we will not fence that side. if L > W, then makes sense to put one of the "length" sides in the riverbank, in that way, we are saving more fence for the other 3 sides.

Then we will have that, the total length of fence used is:

12km = 2*W + L (two times the width and only one time the length are fenced).

From this, we can isolate one of the variables:

L = 12km - 2*W

And replace that in the area equation:

A(W) = (12km - 2*W)*W = 12km*W - 2*W^2.

To maximize this function, we must see when the first derivate is equal to zero:

A'(W) = 12km - 4*W = 0

            12km = 4*W

             12km/4 = 3km = W

Then the width that maximizes the area is W = 3km.

And the maximum area will be:

A'(3km) = 12km*3km - 2*(3km)^2 = 18km^2

And the length can be computed with the equation:

L = 12km - 2*W = 12km - 2*3km = 6km