A country has 10 billion dollars in paper currency in circulation, and each day

Question

A country has 10 billion dollars in paper currency in circulation, and each day 34 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 bank. Let x=x(t) denote the number of new dollars in circulation after t days with units in billions and x(0)=0

A. Determine a differential equation which describes the rate at which x is growing:
dx/dt= ?

B. Solve the differential equation subject to the initial conditions given above
x(t)=?

C. How many days will it take for the new bills to account for 90 percent of the currency in circulation?

Explanation / Answer

The money coming into the banks is exactly proportional to the kind of money(old or new) in circulation,meaning if a proportion of the circulated 10 billions is new(or old),then the same proportion of the 34 millions dollars coming into banks is new(or old) Now,let the amount of old currency at any time t is (10 - x) For 90% in circulation,proportion is (10-x)/10 multiplying this by 34 million to get amount of old currency coming into banks at any time t dx/dt=(.034)(10-x)/10 separate variables dx/(10-x)=.0034dt integrating, -> -ln(10-x)=.0034t apply limits to this.comment for any further help.happy to help

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