Final answer:
The time to double an investment compounded continuously at an annual rate of 6% is calculated using the formula t = ln(2) / r, or estimated using the rule of 70 by dividing 70 by the annual interest rate.
Step-by-step explanation:
The question deals with calculating the time it takes for an investment to double when compounded continuously. An important concept used in addressing this problem is the rule of 70, which is a simplified way to estimate the doubling time of an investment based on its annual growth rate. The rule states that by dividing 70 by the annual interest rate, you can find the approximate number of years it will take for the invested amount to double.
To calculate the exact doubling time for an investment with continuous compounding, we can use the formula involving natural logarithms: t = ln(2) / r, where t is the time to double, ln represents the natural logarithm, and r is the annual interest rate expressed as a decimal. For an interest rate of 6%, or 0.06, the calculation would be t = ln(2) / 0.06. Performing this calculation gives us the exact time it will take for the initial investment to double.
If you prefer a quick approximation without using natural logarithms, you can apply the rule of 70 directly by dividing 70 by 6 (the interest rate). This will give you a close estimate but not the precise time as calculated with continuous compounding. Either way, these methods highlight the power of compound interest and how it can significantly increase an investment over time.