Short answer: Because God took care of Noah and everything else on the Ark.
Bible reference: Genesis 8:1
“And God remembered Noah, and every living thing, and all the cattle that was with him in the ark: and God made a wind to pass over the earth, and the waters assuaged”
The Bible says that “God remembered Noah” and those aboard the Ark during the Flood. When we think of remembering, we usually think of something like not forgetting, but the word “remembered” has a special meaning here. It means that God was thinking of, watching over and taking care of the Ark the entire time.
What’s this have to do with hungry meat-eating dinosaurs and man-eating tigers?
Well, God sent all of the animals to Noah and commanded him to bring them on board. This verse tells us that He took care of them. We can only guess how God kept everyone safe on the Ark during that voyage – including safe from each other! But the Biblle does give us some clues.
For example, the Book of Daniel records that God’s prophet was thrown into a den of hungry lions, but he wasn’t eaten because because God locked the jaws of the lions. By His power, He kept them from eating Daniel.
The Book of Genesis also tells us that God originally created the animals to eat plants. A lot of animals have sharp teeth, but sharp teeth doesn’t always mean that animal eats meat. Pandas, mountain gorillas, lemurs and fruit bats have big sharp teeth, but they don’t eat meat as a rule. also, you may have a pet, such as a dog or a cat. They do eat meat, but you may have noticed that sometimes they eat grass. creatiuon scientists believe that God may have caused meat eaters to eat plants again like they were originally created to do, before Adam sinned and the world was cursed. After they left the Ark, God would have stopped suppressing their fallen urge to eat meat and let them return to being predators.
We could also add that when animals go through a disaster, such as a forest fire or earthquake, they go into a state of near hibernation. And many animals hibernate [or sleep for a season] and do not need to eat during this time. It’s possible that many animals, including meat eaters like lions and T-rexes could have gone into hibernation for long periods of the Ark’s voyage.
While the Bible doesn’t tell us the details of how God “remembered” those on the Ark, there are reasonable answers if we think about it.
Noah's Ark: Beyond Flannelgraph 
Genesis 9:1-2: And God blessed Noah and his sons, and said unto them, Be fruitful, and multiply, and replenish the earth. And the fear of you and dread of you shall be upon every fowl of the air, upon all that moveth [upon] the earth, and upon all the fishes of the sea; into your hand are they delivered.
This is when the separation of peace between man and animal ended and is now what you see today and the reason why they lived in harmony while in the ark.
Brian,
While your remarks explain why lions and tigers and T-rexes didn’t eat Noah and his family, this verse doesn’t explain why the animals got along. In fact, if the fossil record is largely a record of the animals that existed at the time of the Flood, it should be pointed out that we do see evidence of animals eating other animals in the fossil record.
Regards,
Tony
It would seem logical that canivores ate other animals before the flood. Genesis 6:12 was explicit when it says all ‘flesh’ were corrupt, I see this as meaning both man and animal because they’ve deviated from the way God intended things to be as a result of the fall. But also genesis9:1-2 explains why the animals did not attack noah and his family. As for the animal attacking each other we also have to account that lions were not probably on the same floor as goats or cows. Quick question though Lions and Wolves, most canivores are pack hunters and two of a kind barely passes as a pack, would it not seem plausible that the animals were scared and confused hence were less prone to attacking other animals?
We can infer that if the fossil record is largely the result of a worldwide Flood then carnivory developed sometime between the Fall and the Flood, as evidenced by fossils demonstrating such carnivory.
The whole story is fake and is comprimised as the population gets smarter. This stupid story only worked on sheeps.
I hate to take a cheap shot, but your syntax seriously causes me to question whether evolutionists are smarter.
The real problem with your argument is the existence of people like me, who grew up in church, rebelled against it and believed what you apparently do for about a decade and then came back to Christian orthodoxy after a careful examination of the facts. Meanwhile evos credulously parrot the all-natural Just-so story they were told to believe in public school. One of us is a sheep, but it’s not me: https://siriusknotts.wordpress.com/2008/09/11/why-i-am-a-creationist-3/
lol this is laughable lions don’t hibernate..a more plausible explanation would be that Noah could’ve gotten many of these animals when they’re young during that time they would’ve have been the dangerous animals they’re known to be
Lions don’t hybernate… naturally.
I do like your suggestion of younger animals. In fact, I suggested it elsewhere in the case of dinosaurs; however, it would be difficult to rule out supernatural intervention in a case where a supernatural Creator has already supernaturally drawn said animals to Noah in the first place.
Greetings. Good stuff here. Kind of arrived at this question when pondering tooth form and the emergence of carnivorism in the biblical timeline. At the moment I am attempting to refute the post-flood carnivorism idea, which I don’t feel is accurate, though it would explain this question quite well. I agree with your conclusion that carnivores existed before the flood. At any rate, was just wondering what your thoughts might be on the Answers From Genesis proposition that dried meats could have been brought upon the ark to feed the carnivores? Just ran across this blog by the way and look forward to reading more of it. Take care.
–Brandon
I think that the idea of dried meat and dried fish, while not specifically mentioned in the Bible, is not prohibited by the text. Furthermore, it makes sense.
The idea that carnivory developed only post-Flood is refuted by the fossil record. If the fossil record is, in essence, a record of all of the pre-Flood world’s flora and fauna, we note that there is evidence of carnivorous dinosaurs beyond mere sharp teeth. http://news.bbc.co.uk/earth/hi/earth_news/newsid_8596000/8596568.stm