A farmer plans to fence three sides of a rectangular pasture
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
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 each cost $6 per meter, while the side parallel to the river costs $4 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 in terms of w and the parallel side length l.
(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 physically meaningful values of w.
(c) Determine all critical numbers of A that lie in the interior of the interval identified in part (b).
(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) State the optimal pasture dimensions, including the lengths of all three fenced sides, and the corresponding maximum area.