Continuously Compounded Interest

Uploaded by TheIntegralCALC on 17.08.2011

Hi, everyone. Welcome back to Today, we’re going to be talking about continuously
compounded interest. And this is an application of differential equations that’s really
similar to the much more common population growth application. So in this particular
situation, we’ve been told that our great-great-great-great grandfather owed a 30-cent fine a hundred
years ago on an overdue library book that we just we just found and the fine has grown
exponentially at 5% annual interest compounded continuously. And we’ve been asked to figure
out how much it would cost us if we wanted to pay the fine today.
So it’s a pretty ridiculous question but nevertheless, this is the formula we’re
going to be using. It tells us the amount of the interest after time t. So it relates
those two things and it’s really similar again to the population growth equation. so
in this case, we’ve been given all of the pieces we need to find out how much the fine 
would be today. So we can calculate the amount after time t and we know that the initial
fine was 30 cents so we can go ahead and plug in 0.3 for a sub zero which is the initial
amount or the amount at time zero. then we’ve got e raised to the rate times time. We know
that the rate is 5 percent, annual interest rate. So we’d know where to plug in 0.05
for r; and we know that t is 100 because the fine was 30 cents 100 years ago and so we
plug in 100 for t. And now all we have to do is plug that into our calculator and we
see that the amount is equal to 44 dollars and 52 cents which is the amount of money
that it would cost us to pay off the fine if we wanted to be good citizens and return
the library book. So that’s it. Really simple. I just really wanted to show you guys how
to use this formula and again really similar to population growth formula, I also did an
example with sales decline. They’re all kind of related. You can use this formula
for lots of different kinds of growth and decay sort of equations. So that’s it. I
hope this helped you and I will see you in the next video. Bye!