A rectangular grazing pen will be built against a straight section of river so that…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A rectangular grazing pen will be built against a straight section of river so that no fencing is needed along the side adjacent to the water. A total of 600 meters of fencing is available for the other three sides.
Let x be the length, in meters, of each of the two sides perpendicular to the river, and let A(x) be the enclosed area (in square meters). Assume 0<x<300.
(a) Express A(x) solely in terms of x. Clearly indicate the domain of this formula.
(b) Find all critical points of A on the open interval (0,300) and determine the value of x that yields the largest possible area. Show all calculus work that leads to your value.
(c) Confirm that the critical point you found in part (b) corresponds to a local maximum of A by analyzing concavity.
(d) State the dimensions of the pen that maximize the area and compute the maximum possible area, including appropriate units.