**Problem Statement:**You have 1200 feet of fencing material and you wish to construct a rectangular dog kennel with 3 runways. What should the dimensions of the rectangle be to have a rectangle of maximum area?**Diagram Explanation:**The diagram below the problem statement illustrates the setup:- A long horizontal rectangle represents the "BARN."- Four vertical lines divide the rectangular area below the barn into three separate sections (referred to as the runways).The task is to determine the optimal dimensions of this rectangular area to maximize the total area covered by the runways, given the limited fencing material. Since the barn is one side of the rectangle, fencing will only be needed for three sides per runway (one long side and the partitioning).

Let length of the rectangle be x and breath be y so area is given by: A=xy. Also 2y has to be used to create boundary between the three runaways so total fencing material required is x+4y=1200 ft. Therefore, x=1200-4ySubstituting x=1200-4y in A=xy: A=1200-4yy and differentiating to find the critical point: ddyA=0ddy1200-4yy=01200-8y=0y=150 ftSince, y=150 ft so from the equation x=1200-4y: x=1200-4×150 =1200-600 =600 ftThus, dimensions of the rectangle is 600 ft by 120 ft.