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?

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Identify the formula for continuous compounding: \( A = P e^{rt} \), where \( A \) is the final amount, \( P \) is the principal (initial investment), \( r \) is the annual interest rate (in decimal form), \( t \) is the time in years, and \( e \) is the base of the natural logarithm.

Substitute the given values into the formula. Since the investment triples, \( A = 3P \), \( t = 5 \), and \( P \) is the initial principal. The equation becomes \( 3P = P e^{5r} \).

Simplify the equation by dividing both sides by \( P \) (assuming \( P \neq 0 \)): \( 3 = e^{5r} \).

Take the natural logarithm (\( \ln \)) of both sides to isolate \( r \): \( \ln(3) = 5r \).

Solve for \( r \) by dividing both sides of the equation by 5: \( r = \frac{\ln(3)}{5} \). This gives the annual rate in decimal form. To express it as a percentage, multiply the result by 100.

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주요 개념

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Continuous Compounding

Continuous compounding refers to the process of earning interest on an investment at every possible moment, rather than at discrete intervals. The formula used for continuous compounding is A = Pe^(rt), where A is the amount of money accumulated after time t, P is the principal amount, r is the annual interest rate, and e is the base of the natural logarithm. This method allows for the maximum growth of an investment over time.

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 continuous compounding, the investment grows exponentially as interest is calculated continuously. This concept is crucial for understanding how investments can increase significantly over a relatively short period, especially when compounded continuously.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is used in continuous compounding to solve for the time or rate when dealing with exponential equations. In this scenario, the natural logarithm helps to isolate the variable r (the annual interest rate) when determining how long it takes for an investment to reach a certain value under continuous compounding.