A person invests \( \$ 8000 \) at \( 4 \% \) interest compounded annually for 3 years and then invests the balance (the \( \$ 8000 \) plus the interest earned) in an account at \( 7 \% \) interest for 10 years. Find the combined value of the investment to the nearest cent after 13 years. (Hint: You need to break this up into two steps/calculations. Be sure to round your balance at the end of the first 3 years to the nearest cent so you can use it in the second set of calculations.)

To determine the combined value of the investment after 13 years, we need to break it into two steps. **Step 1:** Calculate the balance after 3 years with the initial investment of \( \$8000 \) at \( 4\% \) interest compounded annually. The formula for compound interest is given by: \[ A = P(1 + r)^t \] 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). - \( t \) is the time the money is invested or borrowed for, in years. Here, \( P = 8000 \), \( r = 0.04 \), and \( t = 3 \): \[ A = 8000(1 + 0.04)^3 = 8000(1.124864) \approx 8998.91 \] So, after 3 years, the balance will be approximately \( \$8998.91 \). **Step 2:** The total amount after the first 3 years is then invested in an account at \( 7\% \) interest for another 10 years. Using the same formula: Here, \( P = 8998.91 \), \( r = 0.07 \), and \( t = 10 \): \[ A = 8998.91(1 + 0.07)^{10} = 8998.91(1.967151) \approx 17606.26 \] Thus, after 13 years, the combined value of the investment is approximately \( \$17606.26 \).