   Chapter 13.4, Problem 13E ### 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

# Suppose that a printing firm considers its production as a continuous income stream. If the annual rate of flow at time t is given by f ( t ) = 97.5 e − 0.2 ( t + 3 ) in thousands of dollars per year and money is worth 6 % compounded continuously, find the present value and future value of the presses over the next 10 years.

To determine

To calculate: The present value and future value of the printing press whose income stream for next 10 years from 6% compounded continuously income stream is approximated by f(t)=97.5e0.2(t+3) thousands dollars per year. Whereby, t is the rate of flow at time yearly.

Explanation

Given Information:

The present value and future value of the printing press whose income stream for next 10 years from 6% compounded continuously income stream is approximated by f(t)=97.5e0.2(t+3) thousands dollars per year. Whereby, t is the rate of flow at 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)=97.5e0.2(t+3)

Since, the income to be calculated for next 10 years:

Thus,

k=10

And

Money is worth 6% compounded continuously,

Thus,

r=0.06

Considering the formula:

Present value=0kf(t)ertdt

Substituting 10 for k, 0.06 for r and 97.5e0.2(t+3) for f(t) to get:

Present Value=01097.5e0.2(t+3).e0.06tdt=97.5010e0.26t0.6dt=[97.50.26e0.26t0.6]010

Solving the limit of the function to get:

Present Value=[97.50.26e0.26(10)0.6][97.50.26e0

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