Suppose that P dollars in principal is invested at an annual...

Suppose that $P$ dollars in principal is invested at an annual interest rate $r$. For interest compounded $n$ times per year, the amount $A(t)$ in the account after $t$ years is given by $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$. If interest is compounded continuously, the amount is given by $A(t)=P e^{n} .$ Suppose an investor deposits $\$ 10,000$ in an account earning $6.0 \%$ interest compounded continuously. Find the total amount in the account for the following time periods. How does the length of time affect the amount of interest earned? a. $5 \mathrm{yr}$ b. $10 \mathrm{yr}$ c. $15 \mathrm{yr}$ d. $20 \mathrm{yr}$ e. $30 \mathrm{yr}$

Suppose that $P$ dollars in principal is invested at an annual interest rate $r$. For interest compounded $n$ times per year, the amount $A(t)$ in the account after $t$ years is given by $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$. If interest is compounded continuously, the amount is given by $A(t)=P e^{n} .$
Suppose an investor deposits $\$ 10,000$ in an account earning $6.0 \%$ interest compounded continuously. Find the total amount in the account for the following time periods. How does the length of time affect the amount of interest earned?
a. $5 \mathrm{yr}$
b. $10 \mathrm{yr}$
c. $15 \mathrm{yr}$
d. $20 \mathrm{yr}$
e. $30 \mathrm{yr}$

Key Concepts

Effect of Time on Interest Accumulation

The length of time plays a critical role in the amount of interest earned under compound interest circumstances. Due to the effect of exponential growth in continuously compounded interest, longer investment periods result in a disproportionately larger amount of accumulated interest, demonstrating the power of investing early and giving money more time to grow.

Exponential Growth

Exponential growth is a process where the quantity increases at a rate proportional to its current value. In the context of continuously compounded interest, the exponential function e^(rt) describes this growth, meaning that even small changes in the rate or time can lead to significantly different amounts accumulated over the long term.

Compound Interest

Compound interest refers to the process where the interest earned on an investment is reinvested so that in future periods, interest is earned on both the principal and the previously accumulated interest. This results in exponential growth rather than linear growth, making it a fundamental concept in finance and investment strategies.

Continuous Compounding

Continuous compounding is the limiting case of compound interest, where the frequency of compounding is increased indefinitely so that interest is effectively added at every possible moment. This concept is mathematically modeled using the exponential function, and the formula A(t) = P · e^(rt) captures how the investment grows continuously with time.

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