On or Off?
Date: 05/26/2001 at 23:38:06 From: Lisa Williams Subject: Maths word problems Our school has an automatic sprinkler system for watering the hockey field overnight. The sprinkler system first switches on at 10:00 p.m. and remains on for a full number of minutes. Then it switches off for three times as many minutes as it is on. It switches off again, on again, and so on throughout the night, repeating the cycle with the same time lengths for each on and off period. Just a few seconds before 10:11 p.m., the security guard notices that the sprinkler system is off; just a few seconds after 1:03 a.m., he notices that it is on. If it is on again at exactly 2:15 a.m., will it be on or off at 3:30 a.m.? I have tried different numbers of minutes, but nothing works. Please show how to work this out.
Date: 05/27/2001 at 16:50:31 From: Doctor Jaffee Subject: Re: Maths word problems Hi Lisa, Here is how I solved this problem. Since the sprinkler is off for three times as long as it is on, during a 24-hour period the sprinkler will be on for exactly 6 hours and off for 18 hours. Since 6 hours is 360 minutes, the amount of time the sprinkler is on during a cycle must be a factor of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, ..., 180, or 360. But we know that at 10:11 p.m. the sprinkler is off, so that eliminates all the numbers greater than 10. A few seconds after 1:03 a.m. the sprinkler is on. 1:03 a.m. is 183 minutes after 10:00 p.m. So, let's check out our possible answers one at a time. If the sprinkler is on for 1 minute and off for 3 minutes, then we could say that one cycle of the sprinkler is 4 minutes. If we divide 183 by 4 we get 45 remainder 3, which means that the sprinkler has gone through 45 complete cycles and is in the 3rd minute of its current cycle. But during the third minute of a four minute cycle the sprinkler is off, so we can eliminate "1 minute on" as a possibility. On the other hand, if the sprinkler is on for 5 minutes and off for 15 minutes, one cycle lasts 20 minutes. If you divide 183 by 20 you get 9 with a remainder of 3 which means that it has completed 9 cycles and is in the 3rd minute of its current cycle. Since the first 5 minutes of the cycle have the water on, we can conclude that the sprinkler will be on at 1:03 a.m. Now 2:15 a.m. is 255 minutes after 10:00 p.m. So, divide 255 by 20 and we get 12 remainder 15 which means that the sprinkler has completed 12 cycles and is in the 15th minute of its current cycle. So, the water should be turned off. We can conclude, then, that we haven't found our solution yet. If you go through the numbers 1, 2, 3, 4, 5, 6, 8, 9, 10 one at a time, however, you will find one that will satisfy all the conditions of the problem. Give it a try and if you want to check your answer with me, write back. If you are having difficulties, let me know and show me what you have done so far, and I'll try to help you some more. Good luck. - Doctor Jaffee, The Math Forum http://mathforum.org/dr.math/
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