Let the shorter side of the rectangle be x and the longer side be y. The perimeter of the rectangle, excluding the side next to the river, will be equal to the total length of fence Joey has. Therefore:
2x + y = 92
y = 92 - 2x
Moreover, the area of the rectangle will be:
Area = xy
Substituting y,
Area = x(92 - 2x)
92x - 2x² = Area
Now, if we differentiate the area with respect to x and equate it to 0, we can find the length that will give us the maximum area.
92 - 4x = 0
x = 23 yards
Therefore, the shorter side should be 23 yards and the longer side should be 46 yards and the total area enclosed will be:
23 * 46 = 1058 square yards