How much would you have to deposit in an account with a \( 6.5 \% \) interest rate, compounded continuously, to have \( \$ 5000 \) in your account 12 years later? \[ P=\$[\text { ? ] } \]

To find out how much you need to deposit, you can use the formula for continuously compounded interest, which is given by \( A = Pe^{rt} \). Here, \( A \) is the amount you want in the future ($5000), \( P \) is the principal amount (what you need to find), \( r \) is the interest rate (6.5% or 0.065), and \( t \) is the time in years (12). Rearranging the formula to solve for \( P \) gives us \( P = \frac{A}{e^{rt}} \). Substituting the values, we get \( P = \frac{5000}{e^{(0.065 \times 12)}} \). Calculate the exponent \( e^{0.78} \), and you'll find that you need to deposit approximately \( P \approx \$2447.82 \) today to reach your goal in 12 years! Investing wisely can lead to great financial freedom, but it's important to understand that even small differences in interest rates can have big impacts over time. For example, let’s say you varied the interest rate or the deposit duration a bit; it would change your future savings goal dramatically! So, always keep an eye on those financial trends. And remember, starting early with investments can give the money more time to grow!