   Chapter 13.4, Problem 15E ### 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

# A 58-year-old couple are considering opening a business of their own. They will either purchase an established Gift and Card Shoppe or open a new Wine Boutique. The Gift Shoppe has a continuous income stream with an annual rate of flow at time t given by G ( t )   =   30 , 000 (dollars per year) and the Wine Boutique has a continuous income stream with a projected annual rate of flow at time t given by W ( t ) =   21 , 600 e 0.08 t (dollars per year) The initial investment is the same for both businesses, and money is worth 10% compounded continuously. Find the present value of each business over the next 7 years (until the couple reach age 65) to see which is the better buy.

To determine

To calculate: The Present Value of the income stream for next 7 years if money is 10% compounded continuously. Where, the Gift and Card Shoppe has income stream of G(t)=30000 dollars per year and Wine Boutique has income stream of W(t)=21,600e0.08t dollars per year. Also compare which business is better buy.

Explanation

Given Information:

The Gift and Card Shoppe has income stream of G(t)=30000 dollars per year and Wine

Boutique has income stream of W(t)=21,600e0.08t dollars per year.

Formula used:

According to the Present Value of a Continuous Income Stream:

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

PV=0kf(t)ertdt.

Calculation:

Consider the income equation of Gift and Card Shoppe,

G(t)=30000

Since, the income is to be calculated for next 7 years,

So, the value of k will be,

k=7

And

Money is worth 10% compounded continuously,

So, the value of r will be,

r=10100=0.1

Consider the formula,

PV=0kf(t)ertdt

Substitute 7 for k, 0.1 for r and 30000 for f(t) and evaluate,

PV=0730000e0.1tdt=[300000e0.1t]07

Now, solve the limit of the function,

PV=[300000e0.1(7)][300000e0.1(0)]=300000(0

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