If \( \$ 2,500 \) is invested at \( 12 \% \) annual interest, which is compounded continuously, what is the account balance after 3 years, assuming no additional deposits or withdrawals are made? a.) \( \$ 3,583.32 \) b.) \( \$ 3,683.46 \) c.) \( \$ 3,512.32 \) d.) \( \$ 3,489.58 \)

To find the account balance for continuous compounding, you can use the formula \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after n years, including interest, \( P \) is the principal amount (the initial amount), \( r \) is the annual interest rate (decimal), and \( t \) is the time the money is invested for in years. In this case: - \( P = 2500 \) - \( r = 0.12 \) - \( t = 3 \) By substituting the values into the formula, we calculate: \[ A = 2500 \cdot e^{0.12 \cdot 3} = 2500 \cdot e^{0.36} \] Using \( e^{0.36} \approx 1.4333 \), \[ A \approx 2500 \cdot 1.4333 \approx 3583.25 \] So, the account balance after 3 years is approximately \( \$3,583.25 \), which rounds to \( \$3,583.32 \). Thus, the correct answer is **a.) \( \$ 3,583.32 \)**. Investing isn't just about picking stocks; it's about understanding how your money grows. Continuous compounding showcases the magic of exponential growth. You see, the more frequently interest is calculated and added, the more you earn. Imagine if you had invested that amount in a savings account or fund that compounds daily – you’d really watch your money blossom! For anyone diving into continuous compounding, be aware of the nuances. One common mistake is forgetting to convert the interest rate into a decimal (e.g., \( 12\% \) becomes \( 0.12 \)). Additionally, always double-check your calculations with the value of \( e \). Misestimation can lead you to think your investment journey will be smoother than it really is—like misreading a treasure map! Keep your arithmetic sharp!