Question

Capital outlay. A company determines that the rate of revenue coming in from a new machine is $R_{1}(t)=8000-100 t$ in dollars per year, for 8 yr, after which the machine will have to be replaced. The company also determines that a different brand of the machine will yield revenue at a rate of $R_{2}(t)=7600-85 t$ a) Find the accumulated present value of the income stream from each machine at an interest rate of $16 \%$ compounded continuously. b) Find the difference in the accumulated present values.

Name: Problem 30, Chapter 5: Calculus and Its Applications
Uploaded: 2020-10-30T03:44:39-08:00
Duration: 3 min 2 s
Channel: Hossam Mohamed

   Capital outlay. A company determines that the rate of revenue coming in from a new machine is $R_{1}(t)=8000-100 t$
in dollars per year, for 8 yr, after which the machine will have to be replaced. The company also determines that a different brand of the machine will yield revenue at a rate of $R_{2}(t)=7600-85 t$
a) Find the accumulated present value of the income stream from each machine at an interest rate of $16 \%$ compounded continuously.
b) Find the difference in the accumulated present values.

Calculus and Its Applications

Marvin L. Bittinger,… 10th Edition

Chapter 5, Problem 30

Instant Answer

Solved by Expert Hossam Mohamed

10/30/2020

Step 1

The present value of a continuous income stream is given by the integral of the income rate function, discounted at the continuous interest rate. For the first machine, the present value $P_{1}$ is given by: \[P_{1} = \int_{0}^{8} R_{1}(t) e^{-0.16t} dt = Show more…

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Capital outlay. A company determines that the rate of revenue coming in from a new machine is $R_{1}(t)=8000-100 t$ in dollars per year, for 8 yr, after which the machine will have to be replaced. The company also determines that a different brand of the machine will yield revenue at a rate of $R_{2}(t)=7600-85 t$ a) Find the accumulated present value of the income stream from each machine at an interest rate of $16 \%$ compounded continuously. b) Find the difference in the accumulated present values.

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Key Concepts

Present Value

This concept refers to the current worth of a future stream of cash flows, calculated by discounting future amounts at an appropriate interest rate. It allows comparison of cash flows received at different times by converting them into today's dollars.

Continuous Compounding

Continuous compounding assumes that interest accrues instantly and continuously, leading to a discount factor expressed as an exponential function. This concept is used to accurately account for the time value of money when cash flows occur continuously over time.

Integration of Cash Flows

In cases where cash flows vary continuously, integration is used to accumulate the present values of infinitesimally small amounts over a period. By integrating the product of the cash flow function and the continuous discount factor, one can obtain the total present value of a revenue stream.

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