The basic approach to the problem is as follows. You equate the amount of heat coming from the inlet during a certain time interval t to the heat accumulated in the warm “sausage” of around the pipe, of unknown length L and of a radius r. You can get clues from Hint 1 about how to relate the radius r to the time t. NB! If you are not careful enough, you can arrive at a (wrong) conclusion that the length L remains always constant.
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