   Chapter 13.4, Problem 14E ### 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 vending machine company is considering selling some of its machines. Suppose further that the income from these particular machines is a continuous stream with an annual rate of flow at time t given by f ( t ) = 12 e − 0.4 ( t + 3 ) in thousands of dollars per year. Find the present value and future value of the machines over the next 5 years if money is worth 10 % compounded continuously.

To determine

To calculate: The present value and future value of the vending machine company whose income stream for next 5 years from 10% compounded continuously income stream is approximated by f(t)=12e0.4(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 vending machine company whose income stream for next 5 years from 10% compounded continuously income stream is approximated by f(t)=12e0.4(t+3) thousands 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)=12e0.4(t+3)

Since, the income to be calculated for next 5 years:

Thus

k=5

And

Money is worth 10% compounded continuously,

Thus,

r=0.10

Considering the formula:

Present value=0kf(t)ertdt

Substituting 5 for k, 0.10 for r and 12e0.4(t+3) for f(t) to get:

Present value=0512e0.4(t+3).e0.10tdt=0512e0.5t1.2dt=240512e0.5t1.2(0.5)dt=[24e0.5t1.2]05

Solving the limit of the function to get:

Present value=[24e0.5(5)1.2][24e0

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