We want to find the accumulated present value of an investment over a ten-year period
if there is a continuous money flow of four thousand eight hundred dollars per
year and the interest rate is two percent compounded continuously.
So, when we find the accumulated present value of a continuous money flow over a period of time,
what we're determining is the one-time investment we would need to make today
to have the same amount in an account as we would
for the continuous money flow over the same period of time.
So for this problem, the present value is going to be equal to the integral from zero to ten, because
it's ten years. r of t is a continuous money flow per year, so that's four thousand eight hundred.
We’re going to have e raised to the power of negative k times t, where k is the interest
rate. So we're going to have negative zero point zero two t dt. To integrate this, we'll have to
perform a u substitution where u is going to be equal to negative zero point zero two t.
So, the differential u is going to be negative zero point zero two dt.
If we solve this for dt, we'll have to divide both sides by negative zero point zero two.
So dt is going to be equal to du divided by negative zero point zero two. So we'll have the
present value. It's going to be equal to, let's go ahead and pull out the four thousand eight hundred
times the integral. We'll leave off the limits of integration
temporarily because these are in terms of t.
So we'll have e to the u,
but then again dt is du divided by negative zero point zero two. So I'll put the du here,
and then we'll divide four thousand eight hundred by the negative zero point zero two.
Well, the antiderivative of e to the u is just e to the u,
which is really e to the negative zero point zero two t.
And now we need to evaluate this at ten and zero.
So we'll have the present value is going to be equal to four thousand eight hundred
divided by negative zero point zero two times e to the negative zero point zero two times ten.
That's going to be negative zero point two minus, we'll have e to the zero, which is equal to one.
Now we'll evaluate this on the calculator.
Going to multiply this by e to the negative zero point two minus e to the zero,
and e to the zero is equal to one, so we'll just put minus one.
If we round this to the nearest penny,
we'll have forty-three thousand five hundred four dollars and sixty-two cents.
To make better sense of what this means, I went ahead and determined the future
value of this continuous cash flow. So the future value of this continuous cash flow
is approximately fifty-three thousand one hundred thirty-six dollars and sixty-six cents.
So the present value tells us that if we want to have the same amount in the account ten years from
now, we'll have to deposit forty-three thousand five hundred four dollars and
sixty-two cents today and leave it alone for ten years to have the same account balance.
I hope you found this example helpful.