How can I mathematically model the flow of new currency into circulation?

[Miagi]

Homework Statement

A small country has $10 billion in paper currency in circulation, and each day $50 million comes into the country's banks. The government decides to introduce currency by having the backs replace old bills with new ones whenever old currency comes into the banks. Let L=L(t) denote the amount of new currency in circulation at time t, with L(0)=0.

A] Formulate a mathematical model in the form of an initial value problem that represents the flow of the new currency into circulation.

B]Solve for the initial value problem

C] How long does it take for new bills to account for 90% of the currency in circulation?

The Attempt at a Solution

I don't know how to set up this equation. and in order to do the follow ups, I need do A first. Can someone help me?

 

[HallsofIvy]

First, I am going to simplify the numbers by letting L(t) be in terms of billions of dollars. That way, there is a total of 10 (billion dollars) currency in circulation. If there are L(t) in new currency, then there must be 10- L in old currency which will be the fraction (10-L(t))/10 of all currency. You are told that 0.05 (billion dollars= 50 million) comes into the banks. Assuming that "new" and "old' currency are proportioned in that as in the entire country, 0.05(10- L(t))/10= (10- L(t))/200 in old currency comes into the bank and is replaced by new currency each day- that is, the new currency increases by that amount:
$$\frac{dL}{dt}= \frac{10- L(t)}{200}$$

 

[Miagi]

This one was tricky. I'll will try to do the rest with the equation you've given me.

Thank you