A country has 10 billion dollars in paper currency in circulation, and each day 40 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.
-Determine a differential equation which describes the rate at which x is growing: dx/dt=
-How many days will it take for the new bills to account for 90 percent of the currency in circulation?

Question

A country has 10 billion dollars in paper currency in circulation, and each day 40 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. 
-Determine a differential equation which describes the rate at which x is growing: dx/dt=
-How many days will it take for the new bills to account for 90 percent of the currency in circulation?

dumbcow · Answer

The rate the new currency grows depends on how much old currency comes into the banks
also the more new currency in circulation has inverse effect on amount of old currency coming into the banks.
Assume the money coming into the banks is exactly proportional to the type of money in circulation
ie. if half the 10 billion is new currency, then you expect half the 40 million to be new currency

Amount of old currency at any time t is (10 - x)
proportion is (10-x)/10
multiply this by 40 million to get amount of old currency coming into banks at any time t
$$\frac{dx}{dt} = (.04) \frac{10-x}{10}$$
separate variables
$$\frac{dx}{10-x} = .004 dt$$
integrate both sides and solve for x

anonymous · Answer

Thank you!!