Final answer:
To maximize the area of the rectangular roller skating rink with boundaries created using a 300-foot length of rope, the dimensions should be 600 feet by 600 feet.
Step-by-step explanation:
The rectangular roller skating rink
Let's assume the length of the rink is L and the width is W.
Since the rink has no boundaries, if we add the length of the rope to all four sides, the resulting dimensions will be (L+2*300) and (W+2*300).
Calculating the area
The area of a rectangle is given by the formula A = length * width.
So, the area of the rink with the boundaries is (L+2*300) * (W+2*300).
Maximizing the area
To maximize the area, we need to find the dimensions that give us the largest possible product (L+2*300) * (W+2*300).
This can be done by finding the critical points of the function, which in this case is a quadratic function.
However, in this particular case, since there are no restrictions on the dimensions of the rink, the maximum area can be obtained by making L and W as large as possible.
Therefore, the dimensions that should be used to maximize the area are (300+300) and (300+300), which simplifies to 600 and 600.