I'm not 100% certain of this answer since the question is intended to use precalc but I can't think of any alternative to a simple semicircle that would further maximize the area.Basic geometry and the isoperimetric inequality state that given a fixed perimeter, a circle will maximize the contained area. Since the farmer has the river as free perimeter, the most logical solution to this question would be to build a semi-circle with the 1,000ft of fence.The resulting area would be half of that of a circle with a 2,000ft circumference. First we solve for the radius:C = 2pi * r
r = C/(2pi)r = 2000/2pir = 318.31A = pi * r^2A = 318,310 sq ftHalf of that is "159,155 sq ft," which should be the solution to this problem. Please independently verify or have a third party confirm this result.