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ok.. hard maths problem... to do with lighthouses
atlease hard for me...
Two lighthouses can be seen from the sea-front at Shoreton. Both lights switch on and off in regular repeating patterns. One is on for 7 seconds and off for 16 seconds, whilst the other is on for 8 seconds and off 23 seconds.
15 seconds ago both lights became visible at precisely the same moment.
In how many seconds from now will both lights next disappear from view together?
The answer is 568 or 583. You decide!
Two lighthouses can be seen from the sea-front at Shoreton. Both lights switch on and off in regular repeating patterns. One is on for 7 seconds and off for 16 seconds, whilst the other is on for 8 seconds and off 23 seconds.
15 seconds ago both lights became visible at precisely the same moment.
In how many seconds from now will both lights next disappear from view together?
The answer is 568 or 583. You decide!
arslan
atlease hard for me...
Two lighthouses can be seen from the sea-front at Shoreton. Both lights switch on and off in regular repeating patterns. One is on for 7 seconds and off for 16 seconds, whilst the other is on for 8 seconds and off 23 seconds.
In how many seconds from now will both lights next disappear from view together?
The answer is 568 or 583. You decide!
Two lighthouses can be seen from the sea-front at Shoreton. Both lights switch on and off in regular repeating patterns. One is on for 7 seconds and off for 16 seconds, whilst the other is on for 8 seconds and off 23 seconds.
In how many seconds from now will both lights next disappear from view together?
The answer is 568 or 583. You decide!
You've not said whether they start together or what...
7 + 16 = 23. The second light is off for exactly the same amount of time as the first takes to complete one whole 'cycle'. So by the time the second light turns back on, the first is at the same stage as it was when the second turned off. So in one full cycle of the second light, the first light's cycle has advanced by 8 seconds. Can you do it now?
hhmm, still not sure, this is what i can do:
these are the off times in secs since they became visible together:
first light: 7, 30, 53, 76, 99 ...
2nd light: 8, 39, 70, 101, 132 ...
so i suppose when the number is the same for both lights, this is the time it takes for both lights to go off, i.e. disappear at the exact moment.
BUT, the answer is 583 sec or something v. close. so far too tedious to use this method. There must be a simple mathematical solution.
these are the off times in secs since they became visible together:
first light: 7, 30, 53, 76, 99 ...
2nd light: 8, 39, 70, 101, 132 ...
so i suppose when the number is the same for both lights, this is the time it takes for both lights to go off, i.e. disappear at the exact moment.
BUT, the answer is 583 sec or something v. close. so far too tedious to use this method. There must be a simple mathematical solution.
bob_54321
is the answer not 613 seconds? (628-15)
how did u guys do it?
anyway, the book says they go off together 583 seconds after they start together, so it would be 583-15.
since u both got it may be that you're correct, but what i wanted to know is the mathematical approach to take because listing the times like i was doing is far too tedious.
I did the "listing the times" approach because I couldn't be bothered to think. I just banged the numbers into a spreadsheet and looked down the list (only took 15 seconds). I wasn't really concentrating though, so I could have made a mistake (I wouldn't have posted if someone else hadn't said the same thing).
We define an interval as the time elapsed between 2 switch-ons. So the first lighthouse has an interval of 23 secs and the second one has an interval of 31 secs.
We can say 31k - 23r = 7 , where k and r are some positive integers.
It means that at the rth interval of the first lighthouse, the dissapearing of light occurs together within this interval (or at the kth interval of the second lighthouse) Try to visualise a bit by means of graphically representing the problems if you still cant see this relationship.
31k - 23r = 7 is a linear diophantine equation, I would say conveniently so, because 31 and 23 is a coprime. Using very simple euclidean algorithm, k=21 and r=28. So 23*r = 23*28 = 644
So the answer should be 644-16 = 628 because you want the moment the light dissapears, so minus 16secs of switch-off.
It's 4:30am, so my solution maybe completely nonsensical, or there might be a more elegant solution. Oh, btw, you can check out how to solve linear diophantine equation, if you didnt know about it, it's very fascinating and easy to be understood.
We can say 31k - 23r = 7 , where k and r are some positive integers.
It means that at the rth interval of the first lighthouse, the dissapearing of light occurs together within this interval (or at the kth interval of the second lighthouse) Try to visualise a bit by means of graphically representing the problems if you still cant see this relationship.
31k - 23r = 7 is a linear diophantine equation, I would say conveniently so, because 31 and 23 is a coprime. Using very simple euclidean algorithm, k=21 and r=28. So 23*r = 23*28 = 644
So the answer should be 644-16 = 628 because you want the moment the light dissapears, so minus 16secs of switch-off.
It's 4:30am, so my solution maybe completely nonsensical, or there might be a more elegant solution. Oh, btw, you can check out how to solve linear diophantine equation, if you didnt know about it, it's very fascinating and easy to be understood.
khaixiang
We define an interval as the time elapsed between 2 switch-ons. So the first lighthouse has an interval of 23 secs and the second one has an interval of 31 secs.
We can say 31k - 23r = 7 , where k and r are some positive integers.
It means that at the rth interval of the first lighthouse, the dissapearing of light occurs together within this interval (or at the kth interval of the second lighthouse) Try to visualise a bit by means of graphically representing the problems if you still cant see this relationship.
31k - 23r = 7 is a linear diophantine equation, I would say conveniently so, because 31 and 23 is a coprime. Using very simple euclidean algorithm, k=21 and r=28. So 23*r = 23*28 = 628
So the answer should be 644-16 = 628 because you want the moment the light dissapears, so minus 16secs of switch-off.
It's 4:30am, so my solution maybe completely nonsensical, or there might be a more elegant solution. Oh, btw, you can check out how to solve linear diophantine equation, if you didnt know about it, it's very fascinating and easy to be understood.
We can say 31k - 23r = 7 , where k and r are some positive integers.
It means that at the rth interval of the first lighthouse, the dissapearing of light occurs together within this interval (or at the kth interval of the second lighthouse) Try to visualise a bit by means of graphically representing the problems if you still cant see this relationship.
31k - 23r = 7 is a linear diophantine equation, I would say conveniently so, because 31 and 23 is a coprime. Using very simple euclidean algorithm, k=21 and r=28. So 23*r = 23*28 = 628
So the answer should be 644-16 = 628 because you want the moment the light dissapears, so minus 16secs of switch-off.
It's 4:30am, so my solution maybe completely nonsensical, or there might be a more elegant solution. Oh, btw, you can check out how to solve linear diophantine equation, if you didnt know about it, it's very fascinating and easy to be understood.
interesting approach, +ive rep coming ur way
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