Final answer:
The value after 8.3 years of a capital of $3000 invested at 3.5% using continuous compounding is approximately $3952.93.
Step-by-step explanation:
To calculate the value of a $3000 capital invested at a 3.5% interest rate with continuous compounding after 8.3 years, the formula A = P * e^(rt) is employed. In this equation, A represents the final amount, P is the principal, e denotes the base of the natural logarithm, r signifies the interest rate, and t represents the time in years.
By substituting the given values into the formula—specifically, A = 3000 * e^(0.035 * 8.3)—the calculation proceeds. Using a calculator to perform the computation, the result is approximately $3952.93. Rounding this figure to the nearest cent, the final value of the investment after 8.3 years is $3952.93.
The utilization of continuous compounding, facilitated by the exponential function in the formula, allows for a more precise determination of the investment's growth over time compared to traditional compounding methods. In this case, the final amount reflects the compounded value of the initial $3000 capital after 8.3 years, taking into account the continuous compounding effect attributed to the interest rate of 3.5%. This mathematical approach provides investors with a means to project and understand the potential growth of their investments under continuous compounding scenarios.