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?

Verified step by step guidance

1. Identify the formula for continuous compounding: $A=P{e}^{rt}$, where $A$ is the amount of money accumulated after time $t$, $P$ is the principal amount, $r$ is the annual interest rate, and $t$ is the time in years.
2. Set up the equation using the given values: $A=3P$, $P=50000$, and $r=0.075$. Substitute these into the formula to get $3P=P{e}^{0.075t}$.
3. Divide both sides of the equation by $P$ to simplify: $3=e^{0.075t}$.
4. Take the natural logarithm of both sides to solve for $t$: $\ln (3)=0.075t$.
5. Solve for $t$ by dividing both sides by $0.075$: $t=\frac{\ln (3)}{0.075}$.

Key Concepts

Continuous Compounding

Continuous compounding refers to the process of calculating interest on an investment or loan where the interest is added to the principal continuously, rather than at discrete intervals. The formula used 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.

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 is often modeled using the continuous compounding formula, which illustrates how investments can grow significantly due to the effect of compounding interest over time.

Natural Logarithm

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. In this context, it helps to isolate the variable t when determining the time required for an investment to reach a certain value.