   Chapter 13.4, Problem 12E ### 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

# The profit from an insurance agency can be considered as a continuous income stream with an annual rate of flow at time t given by f ( t )   =   840 , 000 (dollars per year). Find the present value and future value of this agency over the next 12 years if money is worth 8 % compounded continuously.

To determine

To calculate: The present value and future value of the profit from an insurance agency income stream for next 12 years from 8% compounded continuously income stream is approximated by f(t)=840,000 dollars per year. Whereby, t is the rate of flow at time yearly.

Explanation

Given Information:

The present value and future value of the profit from an insurance agency income stream for next 12 years from 8% compounded continuously income stream is approximated by f(t)=840,000 dollars per year. Whereby, t is the rate of flow at time yearly.

Formula used:

According to the present value of a continuous income stream:

If f(t) is the continuous income flow earning interest at rate compounded continuously with t=0 to t=k is the time interval, then the Present Value of a continuous income stream is:

Present value=0kf(t)ertdt.

And

According to the future value of a continuous income stream:

If f(t) is the rate of continuous income flow for k years earning interest at rate r compounded continuously, then the present value of a continuous income stream is:

Future value=erk0kf(t)ertdt.

Calculation:

Consider the income equation:

f(t)=840,000

Since, the income to be calculated for next 12 years:

Thus,

k=12

And

Money is worth 8% compounded continuously,

Thus,

r=8100=0.08

Considering the formula:

Present Value=0kf(t)ertdt

Substituting 12 for k, 0.08 for r and 840,000 for f(t) to get:

Present Value=012840,000e0.08tdt=10,500,000012e0.08t(0.08)dt=[10,500,000e0.08t]012

Solving the limit of the function to get:

Present Value=[10,500,000e0

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