Hi guys,
As you can probably see, I am new here. I come to you in desperation, though. My professor has presented us with this problem in which I have no clue where to begin. If somebody could please start me off on the right track, I would greatly appreciate it. Here is the question:
Dear Calculus Student,
I am in a desperate situation, and you are my last resort. When I asked my dear friend Bilbo Baggins for help, he referred me to you, after some excellent help you provided him with redecorating his house. Alas, my predicament is not nearly so domestic.
I, with a small band of dwarves, have recently reclaimed the Mines of Moria from the orcs and goblins who had overtaken it many years ago. Now we are settling into the mines, but the goblins and orcs have not left us. The orcs in particular are acting strangely. I have noticed, through careful observation, that every twenty days a certain cycle repeats: on the day that we firstreclaimedMoria, there were 150 orcs still alive and living in the mountains around us. Then, tendayslater, therewere only 30. Ten days after that, there were 150 again. This pattern has been repeating for as long as I have been here to observe. I might assume they are leaving according to some sort of festival calendar, but I have never seen orcs behave like this. All I know is that this pattern does not seem likely to stop any time soon.
Then there are the goblins. On the day we overtook the mines (that day when 150 orcs were still wandering about), there were 1500 goblins. Two days after that, one of them had left (we saw him sneaking away). Two days later, two of them were seen leaving, and two days after that (that is, on day 6 post-reclamation), three disappeared. The rate of disappearance has been following this pattern ever since.
Here’s where you come in: I mentioned before that we were a small band of dwarves, and we are smaller still since some were lost in the fighting, and others have gone exploring in the mines and not returned. With the strength we have left, we feel we could drive off the remaining orcs and goblins, but only if there were fewer than 40 total creatures left in the mountains around us. I do not have much of a head for figures, so I need you to tell me: will there ever be a day when so few creatures are left? If that day will come, and you can tell us when it will happen, we can be prepared to attack. The incomprehensible behavior of the orcs makes me somehat pessimistic, though. Perhaps we should come up with a different plan to drive them off.
Sincerely,
Balin, son of Fundin, Lord of Moria (for now!)
A note of help:
• It appears that the orc population is following some sort of sine or cosine pattern.
• While he gave you the actual orc population, Balin only gave you the rate of change of the goblin population....
As you can probably see, I am new here. I come to you in desperation, though. My professor has presented us with this problem in which I have no clue where to begin. If somebody could please start me off on the right track, I would greatly appreciate it. Here is the question:
Dear Calculus Student,
I am in a desperate situation, and you are my last resort. When I asked my dear friend Bilbo Baggins for help, he referred me to you, after some excellent help you provided him with redecorating his house. Alas, my predicament is not nearly so domestic.
I, with a small band of dwarves, have recently reclaimed the Mines of Moria from the orcs and goblins who had overtaken it many years ago. Now we are settling into the mines, but the goblins and orcs have not left us. The orcs in particular are acting strangely. I have noticed, through careful observation, that every twenty days a certain cycle repeats: on the day that we firstreclaimedMoria, there were 150 orcs still alive and living in the mountains around us. Then, tendayslater, therewere only 30. Ten days after that, there were 150 again. This pattern has been repeating for as long as I have been here to observe. I might assume they are leaving according to some sort of festival calendar, but I have never seen orcs behave like this. All I know is that this pattern does not seem likely to stop any time soon.
Then there are the goblins. On the day we overtook the mines (that day when 150 orcs were still wandering about), there were 1500 goblins. Two days after that, one of them had left (we saw him sneaking away). Two days later, two of them were seen leaving, and two days after that (that is, on day 6 post-reclamation), three disappeared. The rate of disappearance has been following this pattern ever since.
Here’s where you come in: I mentioned before that we were a small band of dwarves, and we are smaller still since some were lost in the fighting, and others have gone exploring in the mines and not returned. With the strength we have left, we feel we could drive off the remaining orcs and goblins, but only if there were fewer than 40 total creatures left in the mountains around us. I do not have much of a head for figures, so I need you to tell me: will there ever be a day when so few creatures are left? If that day will come, and you can tell us when it will happen, we can be prepared to attack. The incomprehensible behavior of the orcs makes me somehat pessimistic, though. Perhaps we should come up with a different plan to drive them off.
Sincerely,
Balin, son of Fundin, Lord of Moria (for now!)
A note of help:
• It appears that the orc population is following some sort of sine or cosine pattern.
• While he gave you the actual orc population, Balin only gave you the rate of change of the goblin population....
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