Final answer:
The question pertains to using calculus optimization techniques to find the dimensions of a rectangular area with a fixed area that needs the least perimeter of fencing. The optimal solution for the outer rectangle that requires the smallest total length of fence is a square with sides of 14.7 meters, leading to a fence requirement of 73.5 meters.
Step-by-step explanation:
The student is trying to determine the dimensions of a rectangular area that will require the smallest total length of fence for a given area that is to be divided into two equal parts by another fence. In this case, the area is 216m², and we aim to minimize the perimeter for such an area.
To solve this problem, we need to use calculus, specifically optimization techniques. Let's label the length of the rectangle as L and the width as W. Since the area is 216m², we have the equation L x W = 216.
The total length of fencing needed will be represented as P, where P = 2L + 3W (two lengths and three widths, because of the division). By substituting W from the area equation into the perimeter equation, we can find the minimum value of P by taking the derivative and setting it to zero.
After solving, the optimal dimensions will be found to require a square shape where both sides are equal, specifically when both L and W are 14.7m, leading to a minimum total fencing requirement of 73.5m.