A farmer has 600 meters of fencing to construct three equal–width rectangular pens…
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A farmer has 600 meters of fencing to construct three equal–width rectangular pens that share two interior fences and use an existing straight river as one side of the overall enclosure, so no fencing is needed along the river.
Let x be the length (parallel to the river) of each rectangular pen and y be the width (perpendicular to the river). The fencing available is used only for the three interior widths and the two exterior widths, plus the single length opposite the river, as shown in the verbal description.
Assume x and y are measured in meters and satisfy all physical constraints of the situation.
(a) Express the total enclosed area A as a function of x alone. Clearly state the domain of this function imposed by the fencing limitation.
(b) Compute dxdA and determine all critical values of x in the domain found in part (a).
(c) Using calculus and appropriate justification, identify the value of x that yields the absolute maximum enclosed area on the domain. Find the corresponding width y and the maximum area.
(d) Describe, in the context of the farm, what your result from part (c) means for the farmer’s construction plan.