Questions: Investing. How many years will it take 4,000 to grow to 6,100 if it is invested at 6% (A) compounded quarterly? (B) compounded continuously? (A) years (Round to two decimal places.) (B) years (Round to two decimal places.)

Solution Steps

To find the time \( t \) it takes for an investment of \( P = 4000 \) to grow to \( A = 6100 \) at an annual interest rate of \( r = 0.06 \) compounded quarterly (\( n = 4 \)), we use the compound interest formula:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Rearranging to solve for \( t \):

\[ t = \frac{\log\left(\frac{A}{P}\right)}{n \cdot \log\left(1 + \frac{r}{n}\right)} \]

Substituting the values:

\[ t = \frac{\log\left(\frac{6100}{4000}\right)}{4 \cdot \log\left(1 + \frac{0.06}{4}\right)} \approx 7.09 \]

For continuous compounding, we use the formula:

\[ A = Pe^{rt} \]

Rearranging to solve for \( t \):

\[ t = \frac{\log\left(\frac{A}{P}\right)}{r} \]

Substituting the values:

\[ t = \frac{\log\left(\frac{6100}{4000}\right)}{0.06} \approx 7.03 \]

Final Answer