\[ R(5) = 5000e^{0.03 \times 5} = 5000e^{0.15} \approx 5000 \times 1.161834 = 5809.17 \]

["Explanation of R(5) = 5000 e^(0.03 × 5) ≈ 5000 × 1.161834 ≈ 5809.17", "Understanding the Exponential Growth Formula: R(5) = 5000 × e^(0.03 × 5)", "Mathematical modeling of growth phenomena often relies on exponential functions, and one common expression in financial or population modeling is R(5) = 5000 × e^(0.03 × 5). This formula represents exponential growth where:", "- 5000 is the initial amount (often interpreted as principal, base population, or initial investment),\n- 0.03 is the growth rate per time period (expressed as 3% per 5 years),\n- 5 indicates a time span of 25 units (e.g., years),\n- e is Euler’s number (~2.71828), the base of natural logarithms.", "### How It’s Calculated: e^(0.15) and the Result", "The exponent in the expression, 0.03 × 5, equals 0.15. Evaluating e raised to 0.15 yields approximately 1.161834. This value arises from the power series expansion of eˣ:", "[\ne^{0.15} \approx 1 + 0.15 + \frac{0.15^2}{2!} + \frac{0.15^3}{3!} + \cdots \approx 1.161834\n]", "Thus, multiplying 5000 by this scalar gives:", "[\nR(5) = 5000 × 1.161834 \approx 5809.17\n]", "### Real-World Applications: Growth Over Time", "This type of calculation appears in:\n- Compound interest: When investment grows at a continuous rate, the future value follows e^(rt).\n- Population dynamics: A species growing at 3% annually compounded continuously—after 5 years, the population increases by ~16.18%.\n- Epidemiology: Tracking infections expanding at a steady exponential rate.", "### Why Is This Formula Useful?", "Using e^(rt) models simplifies continuous compounding and exact tracking over time. While actual growth may vary (due to resource limits or external shocks), exponential models provide a strong baseline for forecasting.", "In summary, R(5) = 5000 e^(0.15) transfers a constant initial value with a 3% growth rate over five periods into a projected final value—≈5809.17—demonstrating how exponential functions model real-world growth efficiently.", "For precise calculations involving exponential growth, leveraging e and accurate exponent evaluation ensures reliable forecasts across finance, biology, and planning domains."]