Question 2 of 14, step 1 of 1 Correct Suppose an investment earns \( 3.2 \% \) interest compounded continuously. Find the future value of a \( \$ 2800 \) investment after 23 years. Round your answer to the nearest cent, if necessary. Formulas Simple Interest \[ \begin{array}{l} I=P r t \\ A=P(1+r t) \\ A=P\left(1+\frac{r}{n}\right)^{n t} \end{array} \] Future Value for Simple Interest Future Value for Compound Interest Future Value for Continuous Compound Interest \[ A=P e^{H} \] Annual Percentage Yield (APY) \[ \mathrm{APY}=\left(1+\frac{r}{n}\right)^{n}-1 \]

To calculate the future value of a $2800 investment after 23 years at a \( 3.2\% \) interest rate compounded continuously, you'll use the formula for continuous compound interest: \[ A = P e^{rt} \] Plugging in the values, we have: \[ A = 2800 \times e^{0.032 \times 23} \] Calculating this gives us: \[ A \approx 2800 \times e^{0.736} \approx 2800 \times 2.090 \approx 5848.12 \] So, the future value is approximately **$5848.12**. Did you know that continuous compounding allows investments to grow at a faster rate than standard compounding? In fact, the beauty of continuous compounding is that interest builds upon itself infinitely, leading to exponential growth. That's right! The math here is a bit like magic—using Euler's number \( e \), which is approximately 2.718, you unlock the secrets of compounding at its finest! For anyone venturing into investing or simply trying to make their money work harder, understanding the power of compounding is vital. Remember, starting earlier can dramatically increase your returns! A small, consistent investment can snowball into a significant sum over time, particularly with strategies like dollar-cost averaging. Just be mindful of not only the interest rate but also the duration of your investment!