A rancher has 500 meters of fencing to enclose a rectangular field that will be…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A rancher has 500 meters of fencing to enclose a rectangular field that will be divided into three adjacent rectangular pens of equal width by placing two interior fences parallel to one of the shorter sides of the field.
Let the length of each pen (perpendicular to the interior fences) be L meters and the width of each pen (measured along the direction of the interior fences) be W meters.
(a) Express the total amount of fencing used in terms of L and W, and state the resulting constraint equation.
(b) Write the total area A enclosed by all three pens in terms of L and W, then use the constraint from part (a) to express A as a function of a single variable.
(c) Find all critical points of the area function from part (b) within the domain that makes physical sense for this problem.
(d) Determine the dimensions L and W that maximize the total area, and calculate that maximum area to the nearest square meter.