Day 2: Optimization Fence

Citation: "Calculus Early Transcendentals, 6th Edition", by James Stewart Section 4.7 Optimization Problem 11: A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence.


Solution: So first of all, we will need a visualization of the fence we're optimizing. The description of the fence can be drawn with three side by side identical vertical lines and two horizontal lines connecting the tops of the vertical lines. Since all the vertical lines are equal distance, call their length x and call the length of the horizontal lines y. The total area of the fence will be:


And hence


The total perimeter, or length of fencing, is given by:





Keep in mind the problem is asking to minimize the cost of the fence, which, we assume, is analogous to minimizing the length of the fence, identified as "P". We can use the Calculus to minimize the function P(x). Finding P'(x):





This, so far, has provided the optimal x value. We can use this to find the optimal y value:



...and voila, we found the optimal dimensions of the fence with given area. 


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