Final answer:
The continuous compound interest formula A = Pe^(rt) is used to calculate the future value with interest compounded continuously, where P is the principal, r is the annual interest rate, t is time in years, and e is Euler's number (~2.71828). To find the compound interest earned, subtract the principal from the future value.
Step-by-step explanation:
The continuous compound interest formula is used to calculate the amount of interest earned when interest is being compounded continuously. The general compound interest formula is A = P(1 + r/n)^(nt), but for continuous compounding, the formula transforms using the mathematical constant e (Euler's number) and is represented as A = Pe^(rt), where:
- A represents the future value of the investment/loan, including interest.
- P represents the principal amount (the initial amount of money).
- r is the annual interest rate (in decimal form).
- t represents the time the money is invested or borrowed for, in years.
The 'e' in the equation is approximately equal to 2.71828. To find the compound interest earned, we would subtract the principal from the future value: Compound Interest = A - P.
To apply this to a real-world example, if you had a principal amount of $1,000 at an annual interest rate of 5% compounded continuously, you would calculate the future value after 3 years as follows:
A = 1000e^(0.05*3) = 1000 * e^(0.15)
This will give you the total amount after interest, and to find the compound interest itself, you subtract the principal:
Compound Interest = A - 1000
This will tell you how much interest was earned over the 3 years.