Answer:
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.
Step-by-step 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.
