Explanation Given Information: The provided values are c = 30 , 000 + 500 t , r = 7 %, t 1 = 6 years , where c is the income (in dollars), t 1 is time in years and r is the inflation rate. Formula used: When c is the continuous income function in dollars per year at the inflation rate r, then the present value over the t 1 years is given by, Present value = ∫ 0 t 1 c ( t ) e − r t d t Integration by parts. When u and v is assumed to be the differentiable functions of x then, ∫ u d v = u v − ∫ v d u Calculation: Consider the values, c = 30 , 000 + 500 t , r = 7 %, t 1 = 6 years . The income function would be given by, c ( t ) = 30 , 000 + 500 t So, apply the formula to calculate the present value. Substitute c ( t ) = 30 , 000 + 500 t , r = 0.07 and t 1 = 6 in the expression ∫ 0 t 1 c ( t ) e − r t d t to calculate the present value as, Present value = ∫ 0 6 ( 30 , 000 + 500 t ) e − 0.07 t d t = ∫ 0 6 30 , 000 e − 0.07 t d t + 500 ∫ 0 6 t e − 0.07 t d t = − 3 , 000 , 000 7 [ e − 0.07 t ] 0 6 + 500 ∫ 0 6 t e − 0.07 t d t = − 3 , 000 , 000 7 ( e − 0.42 − 1 ) + 500 ∫ 0 6 t e − 0.07 t d t In order to solve the integral ∫ t e − 0.07 t d t apply integration by parts formula, Let u = t and d v = e − 0.07 d t , then d v = e − 0.07 d t Apply integral on both sides of the above equation as, ∫ d v = ∫ e − 0.07 d t v = − 100 7 e − 0.07 t Thus, v = − 100 7 e − 0.07 t , Then differentiate both sides of the equation u = t as, d u = d ( t ) d u = d t So, d u = d t Now apply integration by parts formula. Substitute u = t , d v = e − 0.07 d t , v = − 100 7 e − 0

Calculus: An Applied Approach (Providence College: MTH 109)

9th Edition

ISBN: 9781285142616

Author: Ron Larson

Publisher: CENGAGE C

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Chapter 6.1, Problem 70E

To determine

To calculate: The present value of the income c (in dollars) over t1 years at the provided inflation rate r, given below as,

c=30,000+500t, r=7%, t1=6 years

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Chapter 6.1, Problem 69E

Chapter 6.1, Problem 71E

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Chapter 6 Solutions

Calculus: An Applied Approach (Providence College: MTH 109)

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Prob. 59RE Ch. 6 - Prob. 60RE Ch. 6 - Endowment In Exercises 61 and 62, determine the...Ch. 6 - Prob. 62RE Ch. 6 - Prob. 63RE Ch. 6 - Prob. 64RE Ch. 6 - Prob. 1TYS Ch. 6 - Prob. 2TYS Ch. 6 - Prob. 3TYS Ch. 6 - Prob. 4TYS Ch. 6 - Prob. 5TYS Ch. 6 - Prob. 6TYS Ch. 6 - Prob. 7TYS Ch. 6 - Prob. 8TYS Ch. 6 - Prob. 9TYS Ch. 6 - Prob. 10TYS Ch. 6 - Prob. 11TYS Ch. 6 - Prob. 12TYS Ch. 6 - Prob. 13TYS Ch. 6 - Prob. 14TYS Ch. 6 - Prob. 15TYS Ch. 6 - Prob. 16TYS Ch. 6 - Prob. 17TYS Ch. 6 - Prob. 18TYS

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The present value of the income c (in dollars) over t 1 years at the provided inflation rate r , given below as, c = 30 , 000 + 500 t , r = 7 %, t 1 = 6 years

Solution Summary: The author calculates the present value of the income c (in dollars) over t_1 years at the provided inflation rate r.

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To calculate: The present value of the income c (in dollars) over t1 years at the provided inflation rate r, given below as,

c=30,000+500t, r=7%, t1=6 years

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Chapter 6 Solutions

The present value of the income c (in dollars) over t 1 years at the provided inflation rate r , given below as, c = 30 , 000 + 500 t , r = 7 %, t 1 = 6 years

Solution Summary: The author calculates the present value of the income c (in dollars) over t_1 years at the provided inflation rate r.

Videos

To calculate: The present value of the income c (in dollars) over t1 years at the provided inflation rate r, given below as, c=30,000+500t, r=7%, t1=6 years

Want to see the full answer?

Chapter 6 Solutions

To calculate: The present value of the income c (in dollars) over t1 years at the provided inflation rate r, given below as,

c=30,000+500t, r=7%, t1=6 years