Questions: How long (in years) would 14,000 have to be invested at 5% compounded continuously to earn 3,099.64 interest?. It would have to be invested for years. (Round to 2 decimal places.)

Solution Steps

To find out how long the money needs to be invested, we can use the formula for continuous compounding interest: \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time in years. We need to solve for \( t \) given that the interest earned is $3,099.64, which means the total amount \( A \) is $14,000 + $3,099.64.

We are given the principal amount \( P = 14000 \), the interest earned \( 3099.64 \), and the interest rate \( r = 0.05 \). The total amount \( A \) after earning interest can be calculated as: \[ A = P + \text{interest\_earned} = 14000 + 3099.64 = 17099.64 \]

The formula for continuous compounding is given by: \[ A = Pe^{rt} \] We need to solve for \( t \). Rearranging the formula gives: \[ t = \frac{\ln\left(\frac{A}{P}\right)}{r} \]

Substituting the known values into the equation: \[ t = \frac{\ln\left(\frac{17099.64}{14000}\right)}{0.05} \]

Calculating the value of \( t \) yields: \[ t \approx 4.000001620803334 \] Rounding this to two decimal places gives: \[ t \approx 4.0 \]

Final Answer