Use the formula for continuous compounding to solve Exercises 84–85. How long, to the nearest tenth of a year, will it take $50,000 to triple in value at an annual rate of 7.5% compounded continuously?

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 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 is often modeled using the exponential function, which reflects how investments grow when interest is compounded continuously. Understanding this concept is crucial for predicting how long it will take for an investment to reach a certain value.

The natural logarithm, denoted as ln, is the logarithm to the base e (approximately 2.71828). It is particularly useful in solving equations involving exponential growth, such as those found in continuous compounding. When determining the time required for an investment to grow to a specific amount, the natural logarithm helps isolate the variable t in the continuous compounding formula.