\( \$ 1000 \) are deposited in an account with a \( 8.5 \% \) interest rate, compounded continuously. What is the balance after 5 years? Begin by entering P, or the principal (initial investment). \( P=\$[?] \)
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To determine the balance after 5 years with continuous compounding, we 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 (initial investment), \( r \) is the annual interest rate (decimal), and \( t \) is the number of years the money is invested or borrowed for. Here, \( P = 1000 \), \( r = 0.085 \), and \( t = 5 \). Now, let's calculate: \[ A = 1000 \cdot e^{0.085 \cdot 5} \] Using \( e \approx 2.71828 \), we first calculate \( 0.085 \cdot 5 = 0.425 \), Then, \( e^{0.425} \approx 1.5288 \). Finally, \( A \approx 1000 \cdot 1.5288 \approx 1528.80 \). So, the balance after 5 years is approximately \( \$1528.80 \).
