(a) Explanation Formula used: Present value of money: “Present value at a time T is P = ∫ 0 T f ( t ) e − r t d t where f ( t ) is the rate of continuous money flow at an interest rate r at time t .” Integration by parts: “If u and v are differentiable functions, then ∫ u d v = u v − ∫ v d u .” Calculation: According to the given data f ( t ) = 2000 t − 150 t 2 , r = 0.08 and T = 10 . Substitute the respective values in the present value formula. P = ∫ 0 10 ( 2000 t − 150 t 2 ) e − 0.08 t d t P = ∫ 0 10 2000 t e − 0.08 t d t − ∫ 0 10 150 t 2 e − 0.08 t d t (1) Solve the integral ∫ 0 10 2000 t e − 0.08 t d t . Let u = 2000 t and d v = e − 0.08 t d t . Then, d u d t = 2000 ( 1 ) d u d t = 2000 d u = 2000 d t ∫ d v = ∫ e − 0.08 t d t v = − e − 0.08 t 0.08 Substitute the corresponding values in the above formula and simplify further. ∫ 2000 t e − 0.08 t d t = 2000 t ( − e − 0.08 t 0.08 ) − ∫ ( − e − 0.08 t 0.08 ) ( 2000 ) d t = 2000 t ( − e − 0.08 t 0.08 ) + 2000 0.08 ∫ e − 0.08 t d t = 2000 t ( − e − 0.08 t 0.08 ) + 2000 0.08 ( − e − 0.08 t 0.08 ) = − 2000 0.08 [ e − 0.08 t ( t + 1 0.08 ) ] Apply the limits and simplify, ∫ 0 10 0.01 t e − 0.08 t d t = − 2000 0.08 [ e − 0.08 t ( t + 1 0.08 ) ] 0 10 = − 2000 0.08 [ e − 0 (b) To determine To find: The accumulated amount of money flow at t = 10 if f ( t ) = 2000 t − 150 t 2 , r = 8 % and T = 10 .

10th Edition

ISBN: 8220102020252

Author: RITCHEY

Publisher: PEARSON

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Chapter 16.3, Problem 14E

(a)

To determine

To find: The present value of the money for the time period of 10 years at 8% compounded interest where the rate of money flow is f(t)=2000t−150t2 .

(b)

To determine

To find: The accumulated amount of money flow at t = 10 if f(t)=2000t−150t2 , r=8% and T=10 .

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Chapter 16.3, Problem 13E

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The present value of the money for the time period of 10 years at 8% compounded interest where the rate of money flow is f ( t ) = 2000 t − 150 t 2 .

Solution Summary: The author explains the present value of money for the time period of 10 years at 8% compounded interest.