Final answer:
To find out how long it will take for $10,300 to grow to $20,000 at a 4.8% continuous interest rate, we use the continuous compounding formula. After rearranging to solve for time, the investment duration can be calculated, demonstrating the exponential growth potential of compound interest.
Step-by-step explanation:
The student's question pertains to determining the amount of time it would take for $10,300 to grow into $20,000 when compounded continuously at a rate of 4.8% interest per year. To answer this, we can use the formula for continuous compounding, which is 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), e is the base of the natural logarithm, and t is the time the money is invested for.
To find the time (t), we rearrange the formula to solve for t, getting t = ln(A/P) / (r). Plugging in the given values, we get t = ln(20,000/10,300) / (0.048). After calculating, this yields the number of years required for the investment to grow to $20,000.
This mathematical process highlights the significance of understanding exponential growth and the power of compound interest over time. The example referenced from another question regarding an initial savings of $3,000 that grows into $44,923 in 40 years at a 7% interest rate also showcases the benefits of long-term investment and how compound interest can significantly increase savings over extended periods.