Questions: Find the time required for an investment of 8600 dollars to grow to 11000 dollars at an interest rate of 4.5% per year, compounded continuously. Round to the nearest hundredth of year. t= years.

Solution Steps

To find the time required for an investment to grow to a certain amount with continuous compounding, we use the formula for continuous compound interest: \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate (as a decimal), and \( t \) is the time in years. We need to solve for \( t \).

1. Substitute the given values into the formula: \( A = 11000 \), \( P = 8600 \), and \( r = 0.045 \).
2. Rearrange the formula to solve for \( t \): \( t = \frac{\ln(A/P)}{r} \).
3. Calculate \( t \) using Python.

We are given the following values:

• Principal amount \( P = 8600 \)
• Future amount \( A = 11000 \)
• Annual interest rate \( r = 0.045 \)

The formula for continuous compounding is given by: \[ A = Pe^{rt} \] To find the time \( t \), we rearrange the formula: \[ t = \frac{\ln\left(\frac{A}{P}\right)}{r} \]

Substituting the known values into the rearranged formula: \[ t = \frac{\ln\left(\frac{11000}{8600}\right)}{0.045} \]

Calculating the value: \[ t \approx 5.469623767531301 \] Rounding to the nearest hundredth: \[ t \approx 5.47 \]

Final Answer