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Problem 44 Easy Difficulty

A certain small country has 10 billion dollars in paper currency in circulation, and each day 50 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 banks. Let $ x = x(t) $ denote the amount of new currency in circulation at time $ t, $ with $ x(0) = 0. $
(a) Formulate a mathematical model in the form of an initial-value problem that represents the "flow" of the new currency into circulation.
(b) Solve the initial-value problem found in part (a).
(c) How long will it take for the new bills to account for $ 90% $ of the currency in circulation?

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Video Transcript

Alright so we have that X. Is the amount of new currency in circulation at time. T. So for the problem for part a it's just a formula or a mathematical model in the form of initial value problems that represents the flow of the new currency into circulation for the given information. So since we have that excess amount of new currency in circulation and time t listening the numbers simpler. And assume that eggs is in terms of billions of dollars. Do you notice that there's an implicit assumption that is that the total amount of paper currency remains constant at 10 billion? So the minimum to the maximum value that we have is 10 billion. Um And for that since there are no new bills because the max is 10 billion T. Equals zero and then we have that X. Of zero equals zero. This is our initial value. Now we must find dx over DT. So um the derivative of that and recall that this is simply the rate of change of the amount of new entering the country. So we're assuming that every day 50 million is going to come into the bank. Um The bank then replaces all those bills with new ones. So when T. Equals zero, The 50 million of new bills replaces the 50 million of old bills. However after that some of that 50 million that comes into the bank is new bills because of this. We must figure out what percent of the currency is old bills. Notice that this problem is slightly different in that we do not have a rate in in a rate out. And said we simply replace all the incoming bills with new bills. So dy over dx the derivative depends on how much of the income money is. Almost now on any given day. The change in new bills is going to be DX over DT equals 50,000 and that would be the percentage of old bills. So right now does this match what we think dxc t. Should be at T. Equals zero. Um Is this matching with what the information we have? So above we have dx DT equals 50,000. Um So then if I multiply that by 100 it would be equal um 50 million. So this matches what we reasoned out. Um Now we must figure out the percentage of old bills in the country. We know that the number of new bills is X. So the percentage of new bills Would be X over 10 billion. Therefore the percentage of old bills in the country is 1 -1/10 billion. Thus we have dx over DT um equals 50,000 50 million. 1- X over 10 billion. And our initial value being zero. So given that we move over to question be and that says to solve the initial value found in part a. Um So this seems reasonable because as t increases X increases slower. So this seems likely because as we have more bills in a new country we end up replacing less old bills with new ones. And then moving on to see Um to find how long it takes new currency to reach 90% of the total, we have to set X. of T. equals .9 times 10 billion, which would equal nine billion. Um And sulfur T. That way. So here I have laid out how you solve for it. Um And for that you would receive approximately 4 60.5. But since this is days we've got to round up. So it will take about 461 days for the new currency to reach 90% of the total mountain circulation.

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Video Transcript

James Stewart

Calculus: Early Transcendentals

Catherine Ross

Anna Marie Vagnozzi

Heather Zimmers

Caleb Elmore

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus: Early Transcendentals

Catherine Ross

Anna Marie Vagnozzi

Heather Zimmers

Caleb Elmore

James StewartCalculus: Early Transcendentals

Catherine Ross

Anna Marie Vagnozzi

Heather Zimmers

Caleb Elmore

James Stewart

Calculus: Early Transcendentals