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Mathematical Applications for the ...

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

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BuyFindarrow_forward

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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Sect-13.1 P-9ESect-13.1 P-10ESect-13.1 P-11ESect-13.1 P-12ESect-13.1 P-13ESect-13.1 P-14ESect-13.1 P-15ESect-13.1 P-16ESect-13.1 P-17ESect-13.1 P-18ESect-13.1 P-19ESect-13.1 P-20ESect-13.1 P-21ESect-13.1 P-22ESect-13.1 P-23ESect-13.1 P-24ESect-13.1 P-25ESect-13.1 P-26ESect-13.1 P-27ESect-13.1 P-28ESect-13.1 P-29ESect-13.1 P-30ESect-13.1 P-31ESect-13.1 P-32ESect-13.1 P-33ESect-13.1 P-34ESect-13.1 P-35ESect-13.1 P-36ESect-13.1 P-37ESect-13.1 P-38ESect-13.1 P-39ESect-13.1 P-40ESect-13.2 P-1CPSect-13.2 P-2CPSect-13.2 P-1ESect-13.2 P-2ESect-13.2 P-3ESect-13.2 P-4ESect-13.2 P-5ESect-13.2 P-6ESect-13.2 P-9ESect-13.2 P-10ESect-13.2 P-11ESect-13.2 P-12ESect-13.2 P-7ESect-13.2 P-8ESect-13.2 P-23ESect-13.2 P-24ESect-13.2 P-13ESect-13.2 P-14ESect-13.2 P-15ESect-13.2 P-16ESect-13.2 P-17ESect-13.2 P-18ESect-13.2 P-19ESect-13.2 P-20ESect-13.2 P-21ESect-13.2 P-22ESect-13.2 P-25ESect-13.2 P-26ESect-13.2 P-27ESect-13.2 P-28ESect-13.2 P-29ESect-13.2 P-30ESect-13.2 P-31ESect-13.2 P-32ESect-13.2 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