Final answer:
To maximize the grazing area with 12 meters of fencing, Mike should create a square enclosure. As a rectangle's area is maximized when it is a square, the ideal dimensions are 3 meters by 3 meters.
Step-by-step explanation:
Mike plans to make a rectangular enclosed space with a total fencing material of 12 meters. To maximize the grazing area for his pet rabbit, we need to determine the dimensions that would give the largest enclosed area.
The perimeter P of a rectangle is given by P = 2l + 2w, where l is the length and w is the width of the rectangle. Since Mike has 12 meters of fencing material:
2l + 2w = 12
To maximize the area A, which is given by A = l * w, we use the property that a rectangle with a fixed perimeter has a maximum area when it is a square. Thus the sides must be equal, which means:
l = w
Substituting into the perimeter equation, we get:
2l + 2l = 12
4l = 12
l = 3 meters
So, the dimensions that will maximize the area are 3 meters by 3 meters, as a square is a special case of a rectangle.