Final answer:
To maximize the area using 146 one-foot wide boards without splitting any boards, John can construct a square fence with sides of 36 feet in length, resulting in a total area of 1296 square feet.
Step-by-step explanation:
John has a total of 146 one-foot wide boards to use for fencing a garden and wants to maximize the area.
To achieve this, John should be looking to construct a shape that provides the largest area for a given perimeter, which is a square in planar geometry.
To form a square, each side should have the same length.
Therefore, if John uses all the 146 boards, he can create a square fence with each side consisting of 146 / 4 = 36.5 boards.
However, since he cannot split the boards and needs a whole number, he would use 36 boards per side.
The maximum area that can be constructed using 36 one-foot boards on each side of the square is 36 feet * 36 feet = 1296 square feet.
Any other rectangular configuration with a perimeter of 146 feet would result in less area than the square.
Thus, the dimensions of the maximized area for the garden plot are 36 feet by 36 feet.