   Chapter 13.7, Problem 39E ### 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 Suppose that a business provides a continuous income stream with an annual rate of flow at time t given by f ( t )   = 120 e 0.04 t in thousands of dollars per year. If the interest rate is 9 % compounded continuously, find the capital value of the business.

To determine

To calculate: The capital value of the business if annual rate of flow at time t is given by f(t)=120e0.04t in thousands of dollars per year and the annual interest rate is 9% compounded continuously.

Explanation

Given Information:

The annual flow rate at time t is f(t)=120e0.04t and the interest rate is 9% compounded continuously.

Formula used:

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

C=0f(t)ertdt

The improper integral in form of a limit of integral is,

af(x)dx=limbabf(x)dx

The exponential rule of integrals,

exdx=ex+C

Calculation:

Consider the provided annual flow rate at time t is, f(t)=120e0.04t.

The provided interest rate is 9% compounded continuously.

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

C=0f(t)ertdt

Substitute 120e0.04t for f(t) and 0.09 for r in above formula to get,

C=0120e0

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