Final answer:
The interest rate, when compounded continuously, that would grow $1000 to $1144.54 in 3 years is approximately 4.65%.
Step-by-step explanation:
To determine the continuous compound interest rate, we use the formula A = Pert, where A is the final amount, P is the principal amount, r is the rate of interest, t is the time in years, and e is the base of the natural logarithms (approximately 2.71828). Given a principal amount of $1000 that grows to $1144.54 over 3 years, we have:
1144.54 = 1000e3k
To find the interest rate k, we divide both sides by 1000:
1.14454 = e3k
Now we take the natural logarithm of both sides to solve for k:
ln(1.14454) = 3k
Finally, we can find k by dividing the natural log of 1.14454 by 3:
k = ln(1.14454) / 3
Therefore, the interest rate is:
k = ln(1.14454) / 3 ≈ 0.0465 or 4.65%