How long will it take for an investment of 4,000 dollars to double if: a. the interest rate is \( 10 \% \) per year, compounded continuously? (Round your answer to three decimal places) b. the interest rate is \( 8.5 \% \) per year, compounded monthly? (Enter your answer as a whole number of years and months. Round months up to the nearest month where interest is earned. Both blanks must be correct to receive credit.)

To find out how long it will take for your investment to double, we can use two different formulas based on the type of compounding. For continuous compounding at a \( 10\% \) interest rate: The formula we use is \( t = \frac{\ln(2)}{r} \), where \( r \) is the interest rate. Plugging in the numbers: \[ t = \frac{\ln(2)}{0.10} \approx 6.931 \] So, it will take approximately \( 6.931 \) years, which rounds to **6.931 years**. For monthly compounding at an \( 8.5\% \) interest rate: The formula we can use is \( t = \frac{\ln(2)}{m \cdot \ln(1 + \frac{r}{m})} \), where \( m \) is the frequency of compounding per year. Here, \( r = 0.085 \) and \( m = 12 \): 1. First, calculate \( \frac{r}{m} = \frac{0.085}{12} \approx 0.0070833 \) 2. Next, find \( \ln(1 + 0.0070833) \approx 0.0070796 \) Now we can substitute these values into the formula: \[ t = \frac{\ln(2)}{12 \cdot 0.0070796} \approx \frac{0.6931}{0.084955} \approx 8.166 \] This means it will take approximately \( 8 \) years and about \( 2 \) months (since we round up for the months). Thus, the final answer is **8 years and 2 months**.