Question

A certain small country has $20 billion in paper currency in circulation and each day $80 million comes into the country's banks. The governmnet decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let x(t) be the amount of new currency in circulation in billions of dollars at any time t (in days) and that initially there was $24 million of new currency in circulation. (a) Find an expression for x(t)= (b) How many days will it take for the new bills to account for 83% of the currency in circulation? Round your answer to one decimal.

Answer 1

(a) The expression for x(t) is given by:

x(t) = 20 + 0.08t - 0.02t²

(b) It will take approximately 13.1 days for the new bills to account for 83% of the currency in circulation.

Step-by-step explanation:

The given scenario involves the introduction of new currency, and we need to find an expression for x(t), the amount of new currency in circulation. Initially, there's $24 million of new currency, and each day $80 million comes into the country's banks. The expression is derived from the initial amount of new currency plus the accumulation of daily inflow and a decreasing term to represent the replacement of old bills. The expression is x(t) = 20 + 0.08t - 0.02t², where t is the time in days.

To find the time it takes for the new bills to account for 83% of the currency in circulation, we set x(t) equal to 83% of the total currency. Solving the equation x(t) = 0.83 × 20 gives us a quadratic equation, and solving for (t) yields approximately 13.1 days. This is the time it takes for the new bills to constitute 83% of the currency.

In summary, the expression x(t) models the amount of new currency, and it takes 13.1 days for the new bills to represent 83% of the total currency in circulation. The quadratic equation captures the dynamics of currency replacement over time.