The Kings are saving up to go on a family vacation in 2 years. They invest \( \$ 3100 \) into an account with an annual interest rate of \( 1.21 \% \) compounded annually. Answer the questions below. Do not round any intermediate computations, and round your final answers to the nearest cent. If necessary, refer to the list of financial formulas. (a) Assuming no withdrawals are made, how much money is in the Kings' account after 2 years? (b) How much interest is earned on the Kings' Investment after 2 years? \( \$ \square \)

To find out how much money is in the Kings' account after 2 years with an investment of \( \$3100 \) at an annual interest rate of \( 1.21\% \) compounded annually, we can use the formula for compound interest: \[ A = P(1 + r)^n \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of years the money is invested. (a) Plugging in the values: \[ P = 3100, \quad r = 0.0121, \quad n = 2 \] \[ A = 3100(1 + 0.0121)^2 \] \[ A = 3100(1.0121)^2 \] \[ A = 3100(1.02446441) \approx 3176.84 \] So after 2 years, the Kings will have approximately \( \$3176.84 \). (b) To find out how much interest is earned, we subtract the principal from the total amount after 2 years: \[ \text{Interest} = A - P \] \[ \text{Interest} = 3176.84 - 3100 \approx 76.84 \] Therefore, the interest earned on the Kings' investment after 2 years is approximately \( \$76.84 \). --- Back in the day, compounded interest was quite a revolutionary concept! The earliest instances of banking trace back to ancient Mesopotamia, where farmers used grain as a form of money. They would borrow seeds, and once the crops had grown, they would pay back a portion plus a little extra—like the early seeds of compound interest flourishing into the financial systems we know today! When calculating compound interest, a common mistake is to confuse the compounding periods. Always ensure you're consistently using yearly rates when the compounding is annual. For other compounding schedules, like quarterly or monthly, the rate will need to be adjusted, which can really change your final amount. Double-check your work, and you can avoid some pesky surprises!