its really up to Cook yes. cwi have no one to get him out so its a test of his skill and emotional capability...his nervous capacity.
he can go all the way to 500. he has the time. it will take at most 2 and half days to bowl Hope, Hope and company out twice..at most. it will take less in reality.
so Cook has 2 and half days to bat. he can safely bat all the way to tea on the third and still have time to win the test