QuestionDecember 15, 2025

Find the accumulated present value of an investment over a 8-year period if there is a continuous money flow of 8,000 per year and the interest rate is 1.4% compounded continuously. square Round your answer to the nearest cent

Find the accumulated present value of an investment over a 8-year period if there is a continuous money flow of 8,000 per year and the interest rate is 1.4% compounded continuously. square Round your answer to the nearest cent
Find the accumulated present value of an investment over a 8-year period if there is a continuous money
flow of 8,000 per year and the interest rate is 1.4%  compounded continuously.
 square 
Round your answer to the nearest cent

Solution
3.9(190 votes)

Answer

60,597.19 Explanation 1. Identify the formula for continuous money flow present value For a continuous flow R over T years at continuous rate r: **PV = \int_{0}^{T} R \cdot e^{-rt} \, dt** 2. Substitute values R = 8000, r = 0.014, T = 8: PV = \int_{0}^{8} 8000 \cdot e^{-0.014 t} \, dt 3. Integrate \int e^{-rt} dt = \frac{-1}{r} e^{-rt}: PV = 8000 \cdot \left[ \frac{-1}{0.014} e^{-0.014t} \right]_{0}^{8} 4. Evaluate PV = \frac{-8000}{0.014} \left( e^{-0.014(8)} - e^{0} \right) = \frac{-8000}{0.014} \left( e^{-0.112} - 1 \right) 5. Numerical computation e^{-0.112} \approx 0.89394 Difference: 0.89394 - 1 = -0.10606 PV \approx \frac{-8000}{0.014} \cdot (-0.10606) PV \approx 571428.57 \cdot 0.10606 \approx 60597.19

Explanation

1. Identify the formula for continuous money flow present value <br /> For a continuous flow $R$ over $T$ years at continuous rate $r$: <br />**$PV = \int_{0}^{T} R \cdot e^{-rt} \, dt$** <br /><br />2. Substitute values <br /> $R = 8000$, $r = 0.014$, $T = 8$: <br />$PV = \int_{0}^{8} 8000 \cdot e^{-0.014 t} \, dt$ <br /><br />3. Integrate <br /> $\int e^{-rt} dt = \frac{-1}{r} e^{-rt}$: <br />$PV = 8000 \cdot \left[ \frac{-1}{0.014} e^{-0.014t} \right]_{0}^{8}$ <br /><br />4. Evaluate <br />$PV = \frac{-8000}{0.014} \left( e^{-0.014(8)} - e^{0} \right)$ <br />$= \frac{-8000}{0.014} \left( e^{-0.112} - 1 \right)$ <br /><br />5. Numerical computation <br />$e^{-0.112} \approx 0.89394$ <br />Difference: $0.89394 - 1 = -0.10606$ <br />$PV \approx \frac{-8000}{0.014} \cdot (-0.10606)$ <br />$PV \approx 571428.57 \cdot 0.10606 \approx 60597.19$
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