A certain 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 new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let x = x(t) denote the amount of new currency (in billions of dollars) in circulation at time t (in days), with x(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.

The initial value problem is:

Where x(0) = 0 is the initial condition.

Step-by-step explanation:

Paper Currency in Circulation at any time=$10 billion

Notice that the total amount of paper currency remains constant.

Let x(in billions of dollars) be the amount of new currency in circulation at time t.

The possible maximum value of x is $10 billion since it cannot be higher than paper currency in circulation.

Initially, there are no new bills.

Therefore: x(0) = 0. t,=0

This is our initial value.

Now we must find the rate (dx/dt)

at which new currency is being introduced into circulation.

Every day, $50 million comes into the bank. The bank then replaces all of those bill with new ones.

At t = 0, the $50 million of new bills replaces $50 million of old bills. However subsequently, some of that $50 million that comes into the bank are new bills. Because of this, we must figure out what percent of the currency is old bills.

So dx/dt depends on how much of the incoming money is old bills. Now on any given day, the change in new bills is given as:

dx/dt = $50,000,000(Percentage of old bills).

Now we must figure out the percentage of old bills in the country.

We know that the number of new bills is x.

So the percentage of new bills in the country is given by the function:

Therefore the percentage of old bills in the country is:

Thus we have that the rate of introduction of new bills into circulation is:

Where x(0) = 0.

This is the initial value problem.

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step-by-step explanation: