For seven months or so.

"And the ark rested in the seventh month, on the seventeenth day of the month, upon the mountains of Ararat." Genesis 8:4

For at least ten months.

"And the waters decreased continually until the tenth month: in the tenth month, on the first day of the month, were the tops of the mountains seen." Genesis 8:5

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Explanation:

The ark rested in the seventh month.

Let me use an illustration to explain how the ark rested then. Imagine that you have a giant pool. In the middle is a huge rock of a height that is a foot below the surface. You start to drain the pool, and a raft happens to be right above the rock. The pool water decreases little by little until the raft is resting on top of the rock. The rest of water around the rock does not drain completely until a few hours later.

In this same way, Noah’s ark landed on a high plane. The water was so high and so slow-draining that the ark could have momentarily passed from one mountaintop to another in Ararat-- without the mountains being seen. Only a few months after the ark was resting upon the mountains was Noah and his family able to see where they had landed(the tops of the mountains being seen).

"And Jacob called unto his sons, and said, Gather yourselves together, that I may tell you that which shall befall you in the last days... Dan shall judge his people, as one of the tribes of Israel. Dan shall be a serpent by the way, an adder in the path, that biteth the horse heels, so that his rider shall fall backward. I have waited for thy salvation, O LORD." Genesis 49: 1, 16-18