Stan likes music, and likes oldies the best. In his collection of music he has a cassette tape of �The Cassettes 20 Greatest Hits�. The amazing thing about the 20 Greatest Hits, is that each hit is exactly 3 minutes long. This fits perfectly on a 60-minute tape.

Songs 1 � 10 are on Side A, while 11 � 20 are on Side B. When listening to the tape, Stan does one strange thing. He listens to the first 5 songs on side A, then flips it to the other side. He then listens to Side B from that point until the tape ends. How many songs does Stan hear?

• The tape is wrapped around two spools that have identical diameters of 1 cm.
• The spool that is receiving the tape rotates at a constant speed.
• The tape, when located all on one spool, measures 5 cm in diameter.

Well, I may not be able to get a solution as fast or as good as Charlie's, but I'll attempt to make a shorter one.  I haven't really read Charlie's solution yet.

So if he has listened to 5 songs, that means that side A of the tape is half read, right?  Wrong!  This only means that half of the time that it takes to read the tape has passed.  There's a small, but crucial difference.

Since half of the time has passed, we need to figure out how much tape was used.  I'll measure the tape in area of the ring around the receiving spool.  When side A is done, spool A will have a ring from 1 to 5 cm radius.  This is an area of 24pi.  When half of side A is done, the ring will be 1 to 3 cm radius, with an area of 8pi.

Then, the cassette is reversed.  There are now 16pi square cm on the new receiving spool.  The radius is from 1 to ��17 cm.  The spool needs to wind 5-��17 cm radius' worth of time.  It takes 30 minutes to wind up 4 cm radius' worth.  Therefore, it will take: (5-��17)*30/4 minutes
=6.576707808 minutes
=2.192235936 songs
...plus the first 5 songs
=7.192235936 songs

An exact number is (5-��17)*5/2+5

In case the square root sign doesn't show up, every time you see "��" it is a square root sign.

I hope that was short enough...

Edit:  I realize now that the spool actually has a diameter of 1 cm.  Fortunately, this does not affect the answer, so I will leave it as it is.

Edited on April 29, 2004, 10:30 pm

Posted by Tristan on 2004-04-27 19:33:05