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Compound Interest (Continuously) - Problem 2
Dealing with compound interest for the equation Pert, so for this example what we're looking at is we're given an initial amount $5000 that we're investing, we're told that we're getting 5% interest rate compounded continuously and we're asked to figure out how long it will take to double.
So let's look at our equation a is equals to Pe to the rt. So we're told that our initial amount is $5000 so that goes in for p, you know that e stays there and our rate in this case is 5% and we don't know time, so that's going to be our variable, we leave in this t. And we're asked how long is it going to take to double, so what that tells us is our ending amount is going to be twice our beginning amount. Beginning amount is $5000, so our ending amount is then going to be 10000.
So now we have an exponential equation to solve. Whenever we have this we need to get our exponential term by itself, e to the .05t in this case so we need to divide by 5000 giving us 2 is equal to e to the .05t, we now need to get our exponent down and one way to do that is by taking the log of both sides. We have a base e so it's really easy for us to take natural log of both sides because we'll actually end up getting rid of the base here. So this turns into a natural log of 2 is equal to .05t natural log of e, remember natural log of e goes away so this becomes 1, so we're left with a simple equation to solve.
Natural log of 2 divided by .05 is equal to t. So this is in calculative form if you want to plug it in to get an actual number, natural log of 2 divided by .05 is equal to 13.86.
So after almost 14 years invested at 5% compounded continuously your money will double. Using Pert to solve a doubling problem.