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To find the length of the rectangular dog run space, we need to subtract twice the width from the total perimeter (40 feet) and divide the remaining length by 2.
Let L be the length of the dog run space.
The width is 8 feet, so the total width is 2 * 8 = <<2*8=16>>16 feet.
The total perimeter is 40 feet, so the remaining length is 40 - 16 = <<40-16=24>>24 feet.
Therefore, the length of the dog run space will be 24/2 = <<24/2=12>>12 feet. Answer: \boxed{12}.