Posted by T.Gracken on October 31, 2002 at 18:07:28:
In Reply to: Length of Sleep posted by RD on October 31, 2002 at 14:28:06:
: A person retires on Saturday night at twelve. After tossing for awhile he fell asleep. He awoke while it was still dark. He examined his watch and found that he had been asleep for twice as long as he had lain awake. After tossing for about the same time as before he fell asleep. The second sleep lasted twice as long as before. He awoke again and an hour elapsed before he dozed off again. This time he slept three times as long as on the first occasion and awoke at 8 a.m. How much sleep had he had?
: Please show step by step. Thank you for your assistance.
let T = time tossing (first try at sleep),
then first sleep = 2T.
time tossing after first sleep is about T again,
and second sleep is 4T [twice as long as first sleep].
time tossing after second sleep is 1 hour.
third sleep is 6T (three times first sleep)
total time is 8 hours (12:00 to 8:00)
total time tossing is (2T + 1) hours
total time sleeping is 12T hours
total time slleping and tossing is [(2T+1) + 12T]
so 14T + 1 = 8
=> 14T = 7
=> T = 1/2
so total time spent sleeping is 12*(1/2) = 6 hours.