Final answer:
To find the maximum area that can be enclosed with 900 yd of fencing for a rectangular corral divided into six pens, we must consider the total fencing for the perimeter and the partitions. Using the relationship between the length and width, we can find that the maximum area occurs when the shape of the enclosed space is closest to a square, factoring in the additional partitions. Precise calculations with the given fencing limit will yield the exact maximum area.
Step-by-step explanation:
The problem at hand involves a rectangular corral that needs to be divided into smaller equal pens using 900 yd of fencing. To determine the maximum area that can be enclosed, we need to consider the total fencing available and the number of divisions required for the smaller pens.
Let's define the rectangle's length as L and the width as W. Since there is a need for six equal-sized smaller pens, there will be five partitions parallel to the width. This means the fencing used for the partitions will be 5W. The total perimeter would be 2L + 2W + 5W = 900 due to the fencing available. Simplifying, we get the equation L + 3.5W = 450.
To maximize the area, L × W, we can use the concept of calculus, but another quicker method is understanding that for a given perimeter, a square (or a shape closest to a square) maximizes the area. Our equation deviates from a square because of the additional partitions. We can derive the relationship between L and W and use derivatives to find the maximum area, or we can plug into the equation L = 450 - 3.5W and realize that the closer L is to W (or equivalently L = 3.5W), the larger the enclosed area. We would solve
the system of equations for L and W to find the dimensions that yield the maximum area. Nonetheless, precise calculations are necessary to determine the exact maximum area available for the described conditions.