Final answer:
To compute the final amount with continuous compounding, the formula A = Pe^{rt} is used. For a principal of $1,000, an interest rate of 5%, and a period of 5 years, the correct final amount is approximately $1,284.03, which is not listed in the provided answer choices. Option B is the closest, but it is incorrect for continuous compounding at the given rate and time.
Step-by-step explanation:
To calculate the final amount of an investment with continuous compounding, we use the formula A = Pert, where P is the principal amount ($1,000), r is the annual interest rate (0.05 for 5% interest), t is the time in years (5 years), and e is the base of the natural logarithm (approximately 2.71828).
We plug the values into the formula to get A = 1000 * e(0.05*5), which simplifies to A = 1000 * e0.25. Using a calculator, we find that e0.25 is approximately 1.284025, so the final amount is A = 1000 * 1.284025 which equals $1,284.03.
However, there is an error in the listed answer choices, as none of them match the result. The closest option is B. $1,276.90, but this does not reflect the correct calculation for continuous compounding over five years at a 5% interest rate.