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

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

#### Solutions

Chapter
Section ### 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=0f(t)ertdt

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

af(x)dx=limbabf(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=0f(t)ertdt

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

C=056,000e0.02te0

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