Explanation Formula used: The capital value is C V = ∫ 0 ∞ Re − r t d t , where r is annual rate of interest and t is time. Definition used: Improper integral: “If f is continuous on [ a , ∞ ) , then ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x . If it exists and finite, then it is converges otherwise diverges.” Theorem used: Fundamental Theorem: “Let f be a continuous function on [ a , b ] . If F is anti-derivative of f on [ a , b ] , then ∫ a b f ( x ) d x = F ( b ) − F ( a ) .” Calculation: The income investment is $6,000 per year. Since the income increases at a rate of $200 per year, the total in come become R ( t ) = 6000 + 200 t , at t years. Substitute the value of R ( t ) = 6000 + 200 t and r = 0.05 in the above formula and compute the capital value. C V = ∫ 0 ∞ ( 6 , 000 + 200 t ) e − 0.05 t d t = lim b → ∞ ∫ 0 b ( 6 , 000 + 200 t ) e − 0.05 t d t ( ∵ Definition used ) = ( 6 , 000 ) lim b → ∞ ∫ 0 b e − 0.05 t d t + ( 200 ) lim b → ∞ ∫ 0 b t e − 0

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ISBN: 8220102020252

Author: RITCHEY

Publisher: PEARSON

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Differential Equations

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Chapter 16.4, Problem 46E

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To find: The capital value at an interest rate of 5% for an investment of $6,000 and the income increases linearly at a rate of $200 per year.

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Chapter 16.4, Problem 45E

Chapter 16.4, Problem 47E

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Chapter 16 Solutions

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Ch. 16.1 - Find xe2xdx.Ch. 16.1 - Find ln2xdx.Ch. 16.1 - Prob. 3YT Ch. 16.1 - Find 1ex2lnxdx.Ch. 16.1 - Prob. 5YT Ch. 16.1 - Prob. 1WE Ch. 16.1 - Find the following. W2.(3x52x3)dx Ch. 16.1 - Find the following. W3.(x23+1x)dx Ch. 16.1 - Prob. 4WE Ch. 16.1 - Use integration by parts to find the integrals in...

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The capital value at an interest rate of 5% for an investment of $6,000 and the income increases linearly at a rate of $200 per year.

Solution Summary: The author calculates the capital value at an interest rate of 5% for an investment of 6,000 per year.

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To find: The capital value at an interest rate of 5% for an investment of $6,000 and the income increases linearly at a rate of $200 per year.

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Chapter 16 Solutions