A farmer owns 900 m of fencing to construct a rectangular pasture that borders a straight river. No fencing is needed along the river side. To reduce erosion, the side of the pasture opposite the river must be reinforced at an additional cost of 5 dollars per metre of fencing, while the remaining two sides perpendicular to the river cost 3 dollars per metre. The farmer can spend at most 3600 dollars on fencing materials. The goal is to enclose the pasture of maximum possible area without exceeding either the 900 m fencing supply or the 3600-dollar budget. Answer each part with complete justification.
(a) Using variables of your choice, write expressions for the total length of fencing required and for the total cost.
(b) Translate the physical and financial limitations into mathematical constraints that define the feasible region for the variables.
(c) Obtain a formula for the enclosed area A as a function of a single variable on an appropriate closed interval.
(d) Determine the dimensions of the pasture that maximize A, and compute that maximum area.
(e) Justify that your result in part (d) satisfies all constraints and represents the absolute maximum over the feasible domain.