Final answer:
The question asks to find the optimal dimensions for a composite rectangular enclosure with a fixed amount of fencing, where one rectangle's area is double the other. Calculus and algebra are used to express the problem in terms of a single variable and find the maximum area.
Step-by-step explanation:
The question involves finding the optimal dimensions for a rectangular enclosure made up of two smaller rectangles with a given total amount of fencing. The larger rectangle has twice the area of the smaller one and they share a common edge. To solve this, we could use calculus to maximize the enclosure's area.
Let's define x as the width of the smaller rectangle, and y as the length shared by both rectangles. The area of the smaller rectangle is then x*y, and the larger rectangle, sharing a common edge y, will have a width of 2x since its area is twice the smaller one, making its area 2xy. The total length of fencing needed to enclose both rectangles is the sum of all their sides: 2y + 3x, which must equal the total available fencing, 3900 meters.
By expressing y in terms of x and substituting back into the area expression, we'd get a function of one variable that we could then differentiate to find the maximum area. However, to fully solve for x and y, more information is needed, such as the ratios of sides or an additional constraint. Since problem-solving involves using algebraic manipulation and calculus to find the maximum area of a geometric figure, the knowledge of perimeters, areas, and optimization is crucial.