= 7. A certain small country has $10 billion in paper currency in circulation, and each day $50 million comes into the c

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7 A Certain Small Country Has 10 Billion In Paper Currency In Circulation And Each Day 50 Million Comes Into The C 1 (347.44 KiB) Viewed 22 times

= 7. 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. Since both old bills and new bills will come into the banks while the new currency is gradually introduced, we will need to solve a differential equation to track the amount of new currency in circulation at a given time. Let x(t) denote the amount of new currency, in billions of $, in circulation after t days. Note x(0) = 0. The amount of old money in circulation at time t is 10 – x(t), so the fraction of old money in circulation at time t is 10-X(t) Thus, the amount of old currency coming into the banks each day (and being turned into new currency!) is 10-X(t) · 0.05 billions of dollars. 10 – 2(t) We've shown that new currency is introduced at the rate -0.05, which simplifies to 0.005(10 – 10 x(t)). - 10 10 - This justifies that x(t) satisfies the differential equation dx -0.005(10 – 2). dt (a) (4 points) Solve the differential equation to find x(t). (b) (1 point) At what time t will new bills make up 90% of the currency in circulation?