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A certain small country has 10 billion dollars in paper currency in circulation, and each day 50 million dollars 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 circulation at time $ t, $ 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.

(b) Solve the initial-value problem found in part (a).

(c) How long will it take for the new bills to account for $ 90% $ of the currency in circulation?

a) $\frac{d x}{d t}=50 \times 10^{6}-\frac{x}{200}$

b) $x=10^{10} \cdot\left(1-e^{-t / 200}\right)$

c) 461 days

Differential Equations

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