A farmer owns 900 meters 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 \5permeteroffencing,whiletheremainingtwosidesperpendiculartotherivercost3 per meter. The farmer can spend at most \$3{,}600 on fencing materials.
The goal is to enclose the pasture of maximum possible area without exceeding either the 900-meter fencing supply or the \$3,600 budget.
Answer each of the following parts 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.