1. **Problem statement:** We are given a continuous income stream with rate of flow $f(t) = 900 e^{-0.02 t}$, an interest rate of 7% compounded continuously, and a time period of 3 years. The future value (FV) of this income stream after 3 years is given as 2919. We need to compute the interest earned. 2. **Formula for future value of continuous income stream with continuous compounding:** $$FV = \int_0^T f(t) e^{r(T - t)} dt$$ where $r$ is the continuous interest rate, $T$ is the total time, and $f(t)$ is the income rate at time $t$. 3. **Substitute given values:** - $f(t) = 900 e^{-0.02 t}$ - $r = 0.07$ - $T = 3$ So, $$FV = \int_0^3 900 e^{-0.02 t} e^{0.07 (3 - t)} dt = \int_0^3 900 e^{-0.02 t} e^{0.21 - 0.07 t} dt$$ 4. **Simplify the exponent:** $$e^{-0.02 t} e^{0.21 - 0.07 t} = e^{0.21 - 0.09 t}$$ 5. **Rewrite the integral:** $$FV = 900 e^{0.21} \int_0^3 e^{-0.09 t} dt$$ 6. **Integrate:** $$\int_0^3 e^{-0.09 t} dt = \left[-\frac{1}{0.09} e^{-0.09 t}\right]_0^3 = -\frac{1}{0.09} (e^{-0.27} - 1) = \frac{1 - e^{-0.27}}{0.09}$$ 7. **Calculate the integral value:** $$1 - e^{-0.27} \approx 1 - 0.7634 = 0.2366$$ So, $$\int_0^3 e^{-0.09 t} dt \approx \frac{0.2366}{0.09} = 2.629$$ 8. **Calculate the future value from the integral:** $$FV = 900 e^{0.21} \times 2.629$$ Calculate $e^{0.21} \approx 1.2337$, so $$FV \approx 900 \times 1.2337 \times 2.629 = 900 \times 3.243 = 2919$$ This matches the given future value, confirming our calculations. 9. **Calculate the total principal (sum of income without interest):** The principal is the total income without interest, which is the integral of $f(t)$ from 0 to 3: $$Principal = \int_0^3 900 e^{-0.02 t} dt = 900 \left[-\frac{1}{0.02} e^{-0.02 t}\right]_0^3 = 900 \times 50 (1 - e^{-0.06})$$ Calculate $e^{-0.06} \approx 0.9418$, so $$Principal = 900 \times 50 \times (1 - 0.9418) = 900 \times 50 \times 0.0582 = 900 \times 2.91 = 2619$$ 10. **Calculate interest earned:** $$Interest = FV - Principal = 2919 - 2619 = 300$$ **Final answer:** The interest earned is **300**.