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Mathematical Applications for the Management, Life, and Social Sciences11th Edition

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Chapter 13.7, Problem 41E
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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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Chapter
Section
Problem 1CP
Problem 2CP
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 33E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
Problem 41E
Problem 42E
Problem 43E
Problem 44E
Problem 45E
Problem 46E
Problem 47E
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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

Capital value A business has a continuous income stream with an annual rate of flow at time t given by f ( t ) = 56 , 000 e 0.02 t

(dollars per year). If the interest rate is 10 % compounded continuously, find the capital value of the business.

To determine

To calculate: The capital value of the business if the annual rate of flow at time t is given by f(t)=56,000e0.02t in thousands of dollars per year and the annual interest rate is 10% compounded continuously.

Explanation

Given Information:

Annual flow rate at time t is given by,

f(t)=56,000e0.02t

And the interest rate is 10% compounded continuously.

Formula used:

The capital value for a continuous income stream f(t) and rate r compounded continuously is given by,

C=∫0∞f(t)e−rtdt

If the limit defining the improper integral is a unique finite number, then integral converges else it diverges,

∫a∞f(x)dx=limb→∞∫abf(x)dx

According to the exponential rule of integrals,

∫exdx=ex+C

Calculation:

It is provided that the annual flow rate at time t is given by,

f(t)=56,000e0.02t

And, the provided interest rate is 10% compounded continuously.

Now, the capital value for a continuous income stream f(t) and rate r compounded continuously is given by,

C=∫0∞f(t)e−rtdt

Now, to obtain the capital value, substitute 56,000e0.02t for f(t) and 0.1 for r in above formula to get,

C=∫0∞56,000e0.02te−0

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