Question: How much will the meadow cost?

With thanks to @sand and @janhardo for their help in solving a variational problem using Maple, I'm presenting a puzzle worth remembering. It was originally published in an IBM competition over 20 years ago under simpler conditions than those presented here, and later posed in a more challenging form in private circles:

A farmer wants to buy a section of a large meadow/pasture for his sheep. He likes a lone, large oak tree on the meadow. This tree is to be the starting and ending point of the fence around the desired section. The fence is exactly 100 meters long. The farmer creates a sketch of the situation. He places the oak tree at the origin of a Cartesian coordinate system with a positive y-axis pointing north and sketches the estimated location of the desired meadow section east of the oak tree. The seller then stipulates that the price per square meter increases linearly eastward by a factor of 1/m³. An agreement is reached, and the farmer now considers the optimal route for the fence to maximize the return on his investment.

The fence route without corners and the value of the purchased meadow must be calculated.