A farmer wishes to fence a rectangular field that will be internally subdivided into…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A farmer wishes to fence a rectangular field that will be internally subdivided into three congruent rectangular pens by adding two partitions parallel to one of the shorter sides, as shown conceptually below:
[Description: The overall rectangle has width w (meters) and length ℓ (meters). Two interior fences, each of length ℓ, run parallel to the shorter side so that the field is divided into three side-by-side smaller rectangles.]
The farmer has 600 meters of fencing available for the outer perimeter and the two interior partitions.
(a) Express the total enclosed area A as a function of a single variable, clearly stating any domain restrictions.
(b) Determine the dimensions of the entire field that maximize the total enclosed area.
(c) Justify that the dimensions found in part (b) yield an absolute maximum for the area within the feasible domain.
(d) Interpret your result in the context of the problem, including the maximum total area (with units).