   Chapter 13.6, Problem 33E ### 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

# Present value Suppose that a machine’s production can be considered as a continuous income stream with annual rate of flow at time t given by f ( t )   =   10 , 000   −   500 t (dollars per year)If money is worth 10 % compounded continuously, find the present value of the machine over the next 5 years.

To determine

To calculate: The present value of the machine over the next 5 years if a continuous income stream with annual rate of flow at time t is given by f(t)=10,000500t and money is worth 10% compounded continuously.

Explanation

Given Information:

A continuous income stream with annual rate of flow at time t is given by f(t)=10,000500t and money is worth 10% compounded continuously.

Formula used:

The present value for a continuous income stream f(t) and rate r compounded continuously for T years is given by,

PV=0Tf(t)ertdt

The formula for integration by parts is given by:

udv=uvvdu,

Calculation:

As it is provided a continuous income stream with annual rate of flow at time t is given by

f(t)=10,000500t

And money is worth 10% compounded continuously.

Now, the present value for a continuous income stream f(t) and rate r compounded continuously for T years is given by,

PV=0Tf(t)ertdt

And money is worth 10% compounded continuously, so substitute 0.1 for r, and as the present value is to be obtained over 5 years, substitute 5 for T and 10,000500t for f(t) to get,

PV=05(10,000500t)e0.1tdt=[100,000e0.1t500te0.1dt]05

Now, use the method of integration by parts by splitting the integrands into two components, thus,

u=t

And

dv=e0

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