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Calculus/Word Problem

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Expert: - 2/23/2011


Question
The 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(t) denote the amount of new currency in circulation
at time t, with x(0) = 0.

Formulate an initial value problem for x(t), and solve it. How long will it take for the new bills to account for 90% of the currency in circulation?

Answer
Note that 50 million out of 10 billion is the same as 50 out of a 10,000,
which is is the same as 1/200.

If this is repaced each day, that means that 199/200 is still out there.

Since we want (199/200)^n to be less than 10%, the cutoff point is when
(199/200)^x = 0.1, ant the value of n would be the next integer up from x.
This is the same as (0.995)^x = 0.1.

Taking the ln() of both sides and remembering by doing this that exponets become multipliers,
we get x[ln(0.995)] = ln(0.1), so x = ln(0.1)/ln(0.995), and this is 459.3647642.
Now we want the next integer higher, and that is n = 460,

So, a few days after a year and three months, the requirement of having 90% of the currency replaced will be met.

Calculus

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