5) Katie deposited \( \$ 2,000 \) into a savings account that paid \( 2.5 \% \) interest, compounded continuously. How many years will it take for Katie to have \( \$ 5000 \) ? *Show equation for credit

To find out how many years it will take for Katie's investment to grow from \( \$2,000 \) to \( \$5,000 \) with continuous compounding, we 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 deposit). - \( r \) is the annual interest rate (decimal). - \( t \) is the time the money is invested for in years. - \( e \) is Euler's number, approximately equal to 2.71828. Here \( P = 2000 \), \( A = 5000 \), and \( r = 0.025 \). Plugging in the values: \[ 5000 = 2000 e^{0.025t} \] To isolate \( t \), divide both sides by \( 2000 \): \[ 2.5 = e^{0.025t} \] Applying the natural logarithm to both sides gives: \[ \ln(2.5) = 0.025t \] Now, solve for \( t \): \[ t = \frac{\ln(2.5)}{0.025} \] Calculate \( t \) to find out how many years it will take for Katie to reach her goal. --- In the world of finance, continuous compounding means that interest is calculated and added to the account balance at every moment, maximizing the investment potential. This concept can lead to significantly faster growth compared to traditional compounding methods, which only calculate interest at specific intervals. Only a few savvy investors know that continuously compounding even a small amount can accumulate to staggering sums over time. This principle isn’t just for personal savings. Businesses also leverage continuous compounding to calculate returns on investments, enhancing strategic financial planning!