Angela deposits \( \$ 3500 \) into an account with an APR of \( 4.5 \% \) for 8 years. Find the future value of the account if interest is compounded continuously. Round your answer to the nearest hundredth. 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 \[ A=P e^{r t} \] Annual Percentage Yield (APY) \[ \mathrm{APY}=\left(1+\frac{r}{n}\right)^{n}-1 \]

To find the future value of the account with continuous compounding, we will use the formula for continuous compound interest: \[ A = P e^{rt} \] Plugging in the values: - \( P = 3500 \) - \( r = 0.045 \) (which is \( 4.5\% \)) - \( t = 8 \) Now, calculating: \[ A = 3500 \cdot e^{0.045 \cdot 8} \] \[ A = 3500 \cdot e^{0.36} \] Using \( e^{0.36} \approx 1.4333 \): \[ A \approx 3500 \cdot 1.4333 \approx 5006.55 \] So, the future value of the account is approximately \( \text{\$5006.55} \). Now here are some extra fun facts: Did you know that continuous compounding reflects the idea that interest is being calculated and added to your principal balance every moment? It’s like your money is partying 24/7, racking up more dollars while you sleep! Also, in the world of finance, understanding how different compounding frequencies affect returns can help you choose the best savings accounts or investment options. The more frequently interest is compounded, the more money you'll have in the end! So, let's say "cheers" to making your money work harder for you!