A farmer plans to fence three sides of a rectangular pasture, using a straight river as the fourth side, which requires no fencing. The two sides perpendicular to the river will each cost \6permeter,whilethesideparalleltotheriverwillcost4 per meter. The farmer has a budget of \$2400 for fencing.
The farmer’s goal is to maximize the enclosed area subject to the cost constraint.
(a) Let w be the length (in meters) of each side perpendicular to the river. Express the total fencing cost C(w) in terms of w.
(b) Using the cost constraint, write an explicit formula for the area A(w) of the pasture as a function of w only, and state the closed interval of w values that are physically meaningful.
(c) Determine all critical numbers of A that lie in the interior of the interval identified in part (b). Show analytical work leading to your result.
(d) Evaluate A at the endpoints of the interval and at each critical number, and identify the value of w that yields the maximum possible area. Compute this maximum area.
(e) Interpret your result in context by stating the optimal pasture dimensions (lengths of all three fenced sides) and the corresponding maximum area, including appropriate units.