Final answer:
Using the continuously compounded interest formula, it takes approximately 21.7 years for an initial investment of $25,000 to grow to $300,000 with an annual interest rate of 11.75% when compounded continuously.
Step-by-step explanation:
The question relates to determining the amount of time it takes for an investment to grow to a certain amount with continuously compounded interest.
We start with the continuously compounded interest formula, A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (in decimal form), and t is the time in years.
To solve for the time (t) it takes for $25,000 to grow to $300,000 at an 11.75% annual interest rate, we set up the equation 300,000 = 25,000 * e0.1175t. To isolate t, we divide both sides by 25,000, getting 12 = e0.1175t, and then apply the natural logarithm (ln) to both sides to get ln(12) = 0.1175t.
Solving for t, we have t = ln(12) / 0.1175.
By calculating the value of t, we find t to be approximately 21.7 years, which is the time needed for the principal to grow from $25,000 to $300,000 at a continuously compounded interest rate of 11.75%.
Thus, the power of compound interest becomes evident, as demonstrated by this example and the fact that a larger principal or longer time can greatly increase the total amount of interest accumulated.