What is the meaning of:
to take a whole day up.
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The expression, "to take up" with a time period means to occupy that time period, or to use it up.
"Cleaning up this mess is going to take up the whole day."
"Almost half of my vacation was taken up by bus rides."
"I wish you wouldn't take up my time with this petty stuff."
But I think it is unusual to separate the word "up" from "take" as it appears in your example. Usually people would rearrange it and say, "to take up a whole day."