A farmer has 600 meters of fencing to enclose a rectangular pasture along a straight river
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A farmer has 600 meters of fencing to enclose a rectangular pasture along a straight river. No fencing is needed along the river side. The farmer also wishes to install one fence segment parallel to the river that divides the pasture into two equal smaller rectangular regions.
(a) Define variables and write expressions for the total length of fencing used and the total area of the pasture.
(b) Using the available fencing, express the area solely in terms of one variable.
(c) Find the pasture dimensions that maximize the enclosed area. Provide exact values and verify that the critical point you obtain yields a global maximum on the feasible domain.
(d) Suppose the farmer later decides to add a second fence segment parallel to the river, dividing the pasture into three equal smaller rectangles while keeping the same 600 meters of fencing. Without repeating the full calculation, predict qualitatively whether the maximum total area would increase, decrease, or remain the same. Justify your prediction with reference to your work in parts (a)–(c).