A rectangular pasture of fixed area is to be fenced along three sides—the two sides…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A rectangular pasture of fixed area 2400 m2 is to be fenced along three sides—the two sides perpendicular to a straight river and the side opposite the river. No fence is required along the river bank. The fencing along the sides perpendicular to the river costs 12 dollars per meter, while the fencing along the side parallel to the river costs 20 dollars per meter.
(a) Let x denote the length of each side perpendicular to the river and y the length of the side parallel to the river. Write the total cost C(x) of the fence in terms of x only, using the area constraint.
(b) Compute C′(x) and identify all critical points for x>0 that satisfy the area requirement.
(c) Verify that the critical point yields the minimum fencing cost. State the fencing dimensions that achieve this minimum and the corresponding minimum cost.
(d) Suppose the cost of fencing along the river side increases by k dollars per meter (with k>0). Briefly describe qualitatively, without re‐computing the full optimization, how the optimal value of x will change as k increases.