Chapter 13.7, Problem 38E

### Mathematical Applications for the ...

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

Chapter
Section

### Mathematical Applications for the ...

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

# Capital value Suppose that a donor wants to provide a cash gift to a hospital that will generate a continuous income stream with an annual rate of flow at time t given by f ( t )   =   $20 , 000 per year. If the annual interest rate is 12 % compounded continuously, find the capital value of this perpetuity. To determine To calculate: The capital value of the perpetuity if annual rate of flow at time t is f(t)=$20,000 per year and the annual interest rate is 12%.

Explanation

Given Information:

The annual flow rate for a continuous stream at time t is given as, f(t)=$20,000 per year and the interest rate is 12% that 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 per year at time t is, f(t)=$20,000.

The provided interest rate is 12% 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 20,000 for f(t) and 0.12 for r in above formula to get,

C=020,000e0

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