How long will it take for $\$ 5000$ to grow to $\$ 40900$ at an interest rate of $6.5 \%$ if the interest is compounded continuously? Round the number of years to the nearest hundredth.
A. 1.27 years
B. 3.23 years
C. 32.33 years
D. $3,233.37$ years

Step 1 ：Given that the principal amount (P) is $5000, the final amount (A) is $40900, and the interest rate (r) is 6.5% or 0.065.

Step 2 ：We use the formula for continuous compounding, which is \(A = P e^{rt}\).

Step 3 ：We rearrange this formula to solve for time (t): \(t = \frac{1}{r} \ln \left( \frac{A}{P} \right)\).

Step 4 ：Substitute the given values into the formula: \(t = \frac{1}{0.065} \ln \left( \frac{40900}{5000} \right)\).

Step 5 ：Solving the equation gives us \(t = 32.33372539407162\).

Step 6 ：Rounding to the nearest hundredth, we get \(t = 32.33\) years.

Step 7 ：Final Answer: The time it will take for $5000 to grow to $40900 at an interest rate of 6.5% if the interest is compounded continuously is approximately \(\boxed{32.33}\) years.