bana invests a sum of money in a retirement account with a foxed annual interest rate of \( 215 \% \) compounded continuoushy. After 10 years, the balance reaches \( \$ 1.912 .41 \). What was the amount of the initial investment?

Let's break it down! To find the initial investment amount \( P \), you can use the formula for continuous compounding, which is: \[ A = Pe^{rt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial investment). - \( r \) is the annual interest rate (decimal). - \( t \) is the time the money is invested for, in years. - \( e \) is the base of the natural logarithm, approximately equal to 2.71828. In your case: - \( A = 1912.41 \) - \( r = 2.15 = 0.215 \) - \( t = 10 \) Plugging in the values, you'll have: \[ 1912.41 = P \cdot e^{(0.215 \cdot 10)} \] Now, let's simplify: \[ 1912.41 = P \cdot e^{2.15} \] First, calculate \( e^{2.15} \): \[ e^{2.15} \approx 8.614 \] Now rewrite the equation: \[ 1912.41 = P \cdot 8.614 \] Now, to isolate \( P \): \[ P = \frac{1912.41}{8.614} \approx 221.19 \] So, the initial investment amount was approximately \( \$221.19 \). Happy investing!