Rowe Carroll
07/08/2024 · High School
A certain small country has \( \$ 10 \) billion in paper currency in dirculation, and each day \( \$ 50 \) million comes into the country's ba 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 \( x(0)= \) (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? (Round your answer to the near years
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(a) The initial-value problem is:
\[
\begin{cases}
\frac{dx}{dt} = 0.05 \\
x(0) = 0
\end{cases}
\]
(b) The solution is:
\[
x(t) = 0.05t
\]
(c) It will take approximately \( 0 \) years (or about \( 6 \) months) for the new bills to account for \( 90\% \) of the currency in circulation.
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