Final answer:
The rancher's problem is a calculus optimization issue where the goal is to determine the dimensions of a rectangle with a given area that would use the least amount of fencing, taking into account an additional dividing fence. Calculus derivatives are used to find the minimum perimeter, which dictates the shortest length of fence required.
Step-by-step explanation:
The question revolves around finding the shortest length of fence a rancher can use to enclose a rectangular field of 3,000,000 square feet and then dividing it in half with another fence parallel to one of its sides. To minimize the amount of fencing needed, we will need to use calculus to find the dimensions of the rectangle that result in the smallest perimeter. This is a calculus optimization problem.
To find the total amount of fencing needed, let's call the length of the rectangle 'L' and the width 'W'. The area of the rectangle must be 3,000,000 square feet, so we have L * W = 3,000,000. Additionally, the total fencing needed includes the perimeter of the rectangle plus an additional width section to divide the rectangle in half, which gives us P = 2L + 3W.
Using the area to express 'W' in terms of 'L' (W = 3,000,000/L), and substituting into the expression for 'P', we get P = 2L + 3(3,000,000/L). To find the minimum, we can take the derivative of 'P' with respect to 'L' and find where that derivative is zero. This process will yield the optimal value for 'L', and hence 'W' can be calculated. Inserting these values back into the expression for 'P' gives us the minimum fencing required.