   Chapter 13, Problem 47RE ### 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

# Income streams Suppose that a machine’s production is considered a continuous income stream with an annual rate of flow at time t given by f ( t ) =   150 e − 0.2 t in thousands of dollars per year. Money is worth 8% compounded continuously.(a) Find the present value of the machines production over the next 5 years.(b) Find the future value of the production 5 years from now.Consumer’s and producer’s surplus The demand function for a product under pure competition is p = 64 − 4 x , and the supply function is p =   x   − 1 , where x is the number of units and p is in dollars. Use these functions in Problems 47 and 48.

(a)

To determine

To calculate: The present value of the machine’s production over the next 5 years if the continuous income stream with an annual flow rate at time t is provided by f(t)=150e0.2t in thousands of dollars per year and the rate is 8% compounded continuously.

Explanation

Given Information:

The machine’s production is considered as continuous income stream of an annual rate of flow at time t is f(t)=150e0.2t (in thousands of dollars per year) and the rate is 8% compounded continuously.

Formula used:

For continuous stream, the present value 0kf(t)ertdt, where r is the interest rate.

The exponential rule of integral is eaxdx=1aeax+C, where C is any constant.

Calculation:

Consider the function,

f(t)=150e0.2t

Initially the value of time t is 0 and over the next 5 years, the value of t becomes 5.

The rate of interest is 8%. So, the value of r is,

r=8100=0.08

The present value is,

05f(t)ertdt=05150e0

(b)

To determine

To calculate: The future value of the production 5 years from now if income stream with an annual flow rate at time t is given by f(t)=150e0.2t thousands of dollars per year and the interest rate is 8% compounded continuously.

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