Chapter 11.2, Problem 90ES

Continuous Compounding In our discussion of compound interest, we used annual, semiannual, monthly, quarterly, and daily compounding periods. When interest is compounded daily, it is compounded 360 times a year. If interest were compounded twice daily, it would be compounded $360\left(2\right)=720$ times a year. If interest were compounded four times a day, it would be compounded $360\left(4\right)=1440$ times a year. Remember that the more frequent the compounding period, the more interest earned on the account. Therefore, if interest is compounded more frequently than daily, an investment will earn even more interest than it would if interest were compounded daily. Some banking institutions advertise continuous compounding, which means that the number of compounding periods per year gets very, very large. When compounding continuously, instead of using the compound amount formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}$, we use the following formula. $A=P{e}^n$

In this formula. A is the compound amount when P dollars are deposited at an annual interest rate of r percent compounded continuously for t years. The number e is approximately equal to 2.7182818.

The number e is found in many real-world applications. It is an irrational number, so its decimal representation never terminates or repeats. Calculators have an $\boxed{e^x}$ key for evaluating exponential expressions in which e is the base.

To calculate the compound amount when $10,000 is invested for 5 years at an interest rate of 10%, compounded continuously, use the formula for continuous compounding. Substitute the following values into the formula: $P=10.000,r=10\%=0.10,$ and $t=5$.

$\begin{array}{l}A=P{e}^{rt}\\ A\approx 10,000{\left(2.7182818\right)}^{0.10\left(5\right)}\\ A\approx 16,487.21\end{array}$

The compound amount is $16.487.21.

In the following exercises, calculate the compound interest when Interest is compounded continuously.

a. $P=\$5000,r=8\%,t=6\text{ years}$

b. $P=\$8000,r=7\%,t=15\text{ years}$

c. $P=\$12,000,r=9\%,t=10\text{ years}$

d. $P=\$7000,r=6\%,t=8\text{ years}$

e. $P=\$3000,r=7.5\%,t=4\text{ years}$

f. $P=\$9000,r=8.6\%,t=5\text{ years}$

Solve the following exercises.

g. Calculate the compound amount when $2500 is deposited in an account earning 11% interest, compounded continuously, for 12 years.

h. What is the future value of $15.000 earning 9.5% interest, compounded continuously, for 7 years?

i. How much interest is earned in 9 years on $6000 deposited in an account paying 10% interest, compounded continuously?

j. $25.000 is deposited for 10 years in an account that earns 8% interest. Calculate the future value of the investment if interest is compounded quarterly and if interest is compounded continuously. How much greater is the future value of the investment when interest is compounded continuously?