A square is the best shape to use to maximize area. The ice rinks should be set up as 3 adjacent squares. Let x = the length of the side of one ice rink. This set up would require 10x: 3 to form the top side, 3 to form the bottom side, 1 to form the left side, 1 to form the right side, and 2 to separate the rinks in the middle.
There are 1500 feet of rink boards. 1500 / 10 = 150.
The dimensions will be as follows:
top: 150 * 3 = 450 feet
bottom: 150 * 3 = 450 feet
left side: 150 feet
right side: 150 feet
This will give a total length of 450 feet and a total width of 150 feet. (450 + 450 + 150 + 150 = 1200 feet). The remaining 300 feet will be used as dividers between the rinks, thus using all 1500 feet.
The three square rinks will each have a length and width of 150 feet, thus having an area of 22,500 square feet each. The total area of the three rinks will be 22,500 * 3 = 67,500 square feet. This is the same result as total length * total width: 450 * 150 = 67,500 square feet.