Initial Value Problem: A small country has 10 billion...

[Idealistic]

A small country has 10 billion dollars in circulation. The bank of the country recieves 50 million dollars every day from society. The government decides to replace old curreny (bills) with new currency when money is brought into the bank. Let x(t) denote the amount of new currency in circulation any given day, t. x(0) = 0.

To formulate a mathematical model to represent the flow of new currency in circulation at time t:

dx/dt = (1/10)(10 - 1)0.05; *numbers are in billions.

Im almost positive this is what the set up was on the board. However, I don't understand where it comes from. I mean, the amount of new currency created is 50 million (0.05) per day. But I do understand why 0.05t is not correct, x(t) should be logorithmic and approach x = 10 as t approaches infinity. I just cant concieve the set up

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[DrMike]

A small country has 10 billion dollars in circulation. The bank of the country recieves 50 million dollars every day from society. The government decides to replace old curreny (bills) with new currency when money is brought into the bank. Let x(t) denote the amount of new currency in circulation any given day, t. x(0) = 0.

I think this is wrong :

dx/dt = (1/10)(10 - 1)0.05; *numbers are in billions.

Think about it this way :

If there are x billion in new notes circulating, when the bank gets its 0.05 billion, how much of this is in new notes? How much is in old notes?

If you can answer these, you'll have the differential equation : dx/dt will be equal to the amount coming in to the bank in old notes, since these old notes are all replaced with new ones.

 

[stapel]

To formulate a mathematical model to represent the flow of new currency in circulation at time t:

dx/dt = (1/10)(10 - 1)0.05; *numbers are in billions.

Im almost positive this is what the set up was on the board. However, I don't understand where it comes from.

I believe you're correct in your belief that there is an error in what was put on the board. Gently ask your instructor, in class, for clarification. I'll bet you're not the only one confused! :wink: