Suppose that Canada has a total of $10 billion in $20 bills in circulation, and each day $40 million

Agohofidov6

Agohofidov6

Answered question

2022-01-20

Suppose that Canada has a total of $10 billion in $20 bills in circulation, and each day $40 million of these $20 bills passes through one bank or another. A new harder-to-forge version of the $20 bill is developed, and the banks replace the old bills with new ones whenever they can. How long does it take for the new bills to reach 90% of the total number of $20 bills in circulation?

Answer & Explanation

poleglit3

poleglit3

Beginner2022-01-20Added 32 answers

Since the tag is differential equations, we use a continuous model. Ten billion in 20 dollar bills is 500 million bills. Of these, 2 million pass daily through a bank.
Let x=x(t) be the number of new bills in circulation after time t. We will find an expression for dxdt.
Every day, the fraction 2500 of bills in circulation passes though a bank. If there are x new bills in circulation, then the number of old bills that pass through the bank is
2500(500000000x).
These are replaced. We conclude that
dxdt=2500(500000000x)
To avoid typing such large numbers, let x=500000000y.
Then dydt=500000000dxdt, and quickly we arrive at
dydt=1250(1y)
Initially, y=0. We want to find out when y reaches 0.9. Not that the above differential equation looks very much like the one we get in Newton's Law of Cooling.
Make the change of variable z=1y. Then z is the proportion of old bills after time t. We get dzdt=1250z
This is the familiar differential equation of exponential decay. It has the solution z=z(0)et250.
We have z(0)=1, and want to find the time t such that z(t)=0.10. So we solve the equation et250=0.1.
Take logarithms to the base e. We get t250=ln(0.1). I prefer to note that ln(110)=ln(10), which tells us that t=250ln(10).
Finally, calculate. We get something like t575.

otoplilp1

otoplilp1

Beginner2022-01-21Added 41 answers

Let's say that on say n the proportion of new bills in the population is 0P(n)1. Then each day a fraction f of all the bills go through the banks. 1P(n) of those will be old bills, which will immediately be replaced with new. Therefore
P(n+1)=P(n)+f(1P(n))
which can be rearranged to
P(n+1)=(1f)P(n)+f
If we solve the homogeneous part, we find that P(n)=A(1f)n for some constant A. Now if we guess that P(n)=c solves the whole equation, we get
c=(1f)c+f
from which you can deduce c=1. We know that initially there are no new bills in circulation (i.e. P(0)=0) so we have
0=1+A(1f)0=1+A
and hence A=1, so the general solution is
P(n)=1(1f)n
The question wants to know when this is equal to 90%. We have f=41071010=0.004 and hence 1f=0.996, so
0.9=10.996n
and hence n=log(0.1)log(0.996)=574.5 days.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?