Which half? Is this an unusual lake whose area does not have two halves?
One half of its area, the inner portion, is covered in 47 days. The other half of its area, the outer portion, is covered in one day, the 48th day.
Yes. your final statement is correct. your last two sentences are the entire answer. mathematically.
in reality the truth is they start at the edges and work in to the center. they also start at many location around the perimeter of the lake and never reach the deep areas.