In "The Tell-Tale Heart," how exactly did the narrator satisfy the questions of the police?
The way in which the narrator, initially at least, manages to satisfy the police when they come knocking on his door is by coming up with a convincing lie which is supported by the appearance of the old man's room and his own calm demeanour. Note how the text describes this process:
I bade the gentlemen welcome. The shriek, I said, was my own in a dream. The old man, I mentioned, was absent in the country. I took my visitors all over the house, I bade them search--search well. I led them, at length, to his chamber. I showed them his treasures, secure, undisturbed.
The way in which the narrator is thus able to convince the police and satisfy them is based on his creation of a plausible explanation for the scream and for the absence of the old man, which is supported by the "secure" and "undisturbed" nature of his treasures. If the old man had been killed, presumably the criminal would have stolen his treasures, so showing the police that they have not been meddled with adds great credibility to his story, because obviously the police would not believe at first that anybody would be insane enough to kill without the desire to gain that person's property.