Final answer:
To maximize the area of two adjacent rectangular gardens with a total of 450 meters of fencing, each side of the garden should be 75 meters, assuming that the two gardens form squares.
Step-by-step explanation:
To maximize the area of two adjacent rectangular gardens with a fixed perimeter represented by the available fencing, we should aim to form squares, as a square has the maximum area for a given perimeter among rectangles. In this case, the gardener has 450 meters of fencing to enclose two adjacent rectangular gardens. To find the dimensions that maximize the area, let's denote one side of the first rectangle as x and the adjoining side (which is also a side of the second rectangle) as y. The total perimeter is then 2x + 3y because one side y is shared between the two rectangles.
Setting up the equation 2x + 3y = 450 allows us to express one variable in terms of the other (e.g., y = (450 - 2x)/3). To maximize the area A, which is A = xy, we can use calculus or know that a square maximizes the enclosed area, thus when x=y, the area is maximized. Substituting the expression for y into A = xy and differentiating with respect to x can help to find the maximum area, or we can simply divide the total perimeter by the number of equal sides in two squares, which gives 75 meters for each side of the square.