Grandfather Clock and 7-Second ChimeDate: 06/06/2002 at 11:06:29 From: Cindy Harris Subject: Word problem If it takes a grandfather clock 7 seconds to chime 7 o'clock, how long will it take the same clock to chime 10 o'clock? (The answer is not 10 seconds.) Date: 06/06/2002 at 14:11:00 From: Doctor Ian Subject: Re: Word problem Hi Cindy, A good picture can help a lot: c r [ ]___[ ]___[ ]___[ ]___[ ]___[ ]___[ ] \_____________________________________/ 7 seconds From the time the first chime starts to the time the last one ends is 7 seconds. Let's say that each chime lasts c seconds, and there is a rest of r seconds between chimes. Add up the chimes and the rests, and we get 7 seconds: 7c + 6r = 7 If we extend this for another three chimes, we get three more rests, so 10c + 9r = ? But we have three unknowns and only two equations. Let's make an assumption: A chime takes no time at all, i.e., c=0. Then 7*0 + 6r = 7 6r = 7 r = 7/6 So ? = 9r = 9(7/6) = 3(7/2) = 21/2 = 10.5 seconds But for fun, let's assume that each chime takes half a second to finish. Then we have 7(1/2) + 6r = 7 7/2 + 6r = 7 6r = 7 - 7/2 6r = 7/2 r = 7/12 Now for 10 chimes, we get ? = 10(1/2) + 9(7/12) = 5 + 3(7/4) = 5 + 21/4 = 5 + 5.25 = 10.25 seconds So the answer appears to depend on the duration of each chime. Can we do this more generally? Let's look at that first equation again. We know that 7c + 6r = 7 6r = 7 - 7c 6r = 7(1 - c) r = (7/6)(1 - c) We can substitute this into the second equation: ? = 10c + 9r = 10c + 9[(7/6)(1 - c)] = 10c + 10.5(1 - c) = 10c + 10.5 - 10.5c = 10.5 + 10c - 10.5c = 10.5 - 0.5c Let's check the cases we already know: c = 0 10.5 - 0.5*0 = 10.5 - 0 = 10.5 (Okay) c = 1/2 10.5 - 0.5*(1/2) = 10.5 - 0.25 = 10.25 (Okay) So if you want to get technical, you can't really answer the question without knowing how long each chime lasts. If you assume that each chime is instantaneous, then what's actually going on is that you're spreading (N-1) rests over N seconds; so the time needed for (N+K) chimes is the time needed for (N+K-1) rests: N seconds ------------ * (N+K-1) rests (N-1) rests If 7 chimes take 7 seconds, 10 chimes will take 7 seconds --------- * 9 rests = 63/6 seconds = 10.5 seconds 6 rests If 5 chimes take 5 seconds, 10 chimes will take 5 seconds --------- * 9 rests = 45/4 seconds = 11.25 seconds 4 rests Does that make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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