Explanation Given: Future income stream is F V ( t ) = 5000 + 500 t over next 10 years at an interest of 2 % compounded continuously. Formula used: The present value of the single future payment described will be: P V = F V ( t ) ⋅ e − r t Here, P V is present value, r is rate of interest and t is time in years. Calculation: Recall the present value formula of the single future payment. P V = F V ( t ) ⋅ e − r t Here, P V is present value, r is rate of interest and t is time in years. The payment is to be made over 10 y e a r s , hence a definite integral is to done from t = 0 to t = 10 P V = ∫ t = 0 t = 10 F V ( t ) ⋅ e − r t d t Substitute 5000 + 500 t for F V ( t ) , 0.02 for r and integrate. P V = ∫ t = 0 t = 10 5000 + 500 t ⋅ e − 0.02 t d t Apply integration by parts and consider u = 5000 + 500 t and d v = e − 0.02 t d t which will provide following. d u = 500 d t and, d v = e − 0.02 t d t t v = e − 0.02 t − 0.02 Consider integration by parts. P V = ∫ t = 0 t = 10 ( 5000 + 500 t ) ⋅ e − 0.02 t d t = v u | t = 0 t = 10 − ∫ t = 0 t = 10 v d u = ( ( 5000 + 500 t ) e − 0

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Chapter 13.6, Problem 35E

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Present value of the income to fund a future income stream of FV(t)=5000+500t over next 10 years at an interest of 2% compounded continuously.

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Chapter 13.6, Problem 34E

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Present value of the income to fund a future income stream of F V ( t ) = 5000 + 500 t over next 10 years at an interest of 2 % compounded continuously.

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Present value of the income to fund a future income stream of FV(t)=5000+500t over next 10 years at an interest of 2% compounded continuously.

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