Use the formula for continuous compounding to solve Exercises 84–85. What annual rate, to the nearest percent, is required for an investment subject to continuous compounding to triple in 5 years?

Verified step by step guidance

1. Identify the formula for continuous compounding: \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after n years, including interest, \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the time in years.
2. Since the investment is to triple, set \( A = 3P \).
3. Substitute \( A = 3P \) and \( t = 5 \) into the formula: \( 3P = Pe^{5r} \).
4. Divide both sides by \( P \) to simplify: \( 3 = e^{5r} \).
5. Take the natural logarithm of both sides to solve for \( r \): \( \ln(3) = 5r \), then solve for \( r \) by dividing both sides by 5: \( r = \frac{\ln(3)}{5} \).

Key Concepts

Continuous Compounding

Continuous compounding refers to the process of earning interest on an investment where the interest is calculated and added to the principal continuously, rather than at discrete intervals. The formula used for continuous compounding is A = Pe^(rt), where A is the amount of money accumulated after n years, P is the principal amount, r is the annual interest rate, t is the time in years, and e is the base of the natural logarithm.

Exponential Growth

Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to rapid increases over time. In the context of finance, this means that as interest is compounded continuously, the total amount grows exponentially, which can significantly increase the value of an investment over time, especially with higher rates and longer durations.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately equal to 2.71828. It is particularly useful in continuous compounding calculations because it allows us to solve for the time or rate in the exponential growth formula. In this context, using the natural logarithm helps to isolate the variable of interest, such as the annual rate required for an investment to reach a certain value.