Question

A country has 10 billion dollars in the paper currency in circulation, and each day 33 million dollars come 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 days with units in billions and x(0) = 0.

a. Determine a differential equation which describes the rate at which is growing:

b. Solve the differential equation subject to the initial conditions given above.

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

Answer 1

a.dx/dt = 0.033 billion/day

b. x = 0.033 billion/day · t

c. It will take 273 days for the new bills to account for 90% of the currency in circulation.

Explanation:

Hi there!

a) x(t) is the number of new dollars after t days.

Every day, the number of new dollars is increasing by 0.033 billion.

Then, the rate at which the new dollars are released into the market can be expressed as follows:

dx/dt = 0.033 billion/day

b) Let´s solve the differential equation by separating variables:

dx/dt = 0.033 billion/day

dx = 0.033 billion/day · dt

Integrating both sides from x0 = 0 to x = x and from t = 0 to t = t:

∫dx = ∫ 0.033 billion/day · dt

x - 0 = 0.033 billion/day · t - 0

x = 0.033 billion/day · t

c. The 90% of 10 billion is 9 billion. Replacing x = 9 and solving the equation for t:

x = 0.033 billion/day · t

9 billion = 0.033 billion/day · t

t = 9 billion / 0.033 billion/day

t = 273 days

It will take 273 days for the new bills to account for 90% of the currency in circulation.