Find The Value Of An Income Stream After 16 Years If R ( T ) = 5500 E − 0.02 T R(t) = 5500 E^{-0.02 T} R ( T ) = 5500 E − 0.02 T Is The Rate Of Flow Of Revenue And The Income Is Deposited At A Rate Of 9 Percent Compounded Continuously. Round Intermediate Answers To Eight Decimal Places And

Mar 11, 2025 by ADMIN 291 views

**Finding the Value of an Income Stream after 16 Years**

Introduction

In this article, we will explore the concept of finding the value of an income stream after a certain period of time. This is a crucial aspect of financial planning, as it allows individuals and businesses to determine the future value of their investments. We will use the given rate of flow of revenue, $R(t) = 5500 e^{-0.02 t}$ , and the income deposited at a rate of 9 percent compounded continuously to find the value of the income stream after 16 years.

Understanding the Rate of Flow of Revenue

The rate of flow of revenue, $R(t)$ , is a function that represents the rate at which revenue is generated over time. In this case, the rate of flow of revenue is given by the equation $R(t) = 5500 e^{-0.02 t}$ . This equation indicates that the rate of flow of revenue decreases exponentially over time, with a decay rate of 2% per year.

Understanding Continuous Compounding

Continuous compounding is a method of calculating interest on an investment where the interest is compounded continuously over time. This means that the interest is added to the principal at a rate that is proportional to the current balance. In this case, the income is deposited at a rate of 9 percent compounded continuously.

Calculating the Future Value of the Income Stream

To calculate the future value of the income stream, we need to use the formula for continuous compounding:

FV = \int_{0}^{T} R(t) e^{r(t-T)} dt

where $FV$ is the future value of the income stream, $R(t)$ is the rate of flow of revenue, $r$ is the interest rate, and $T$ is the time period.

Applying the Formula

We can now apply the formula to calculate the future value of the income stream. We will use the given rate of flow of revenue, $R(t) = 5500 e^{-0.02 t}$ , and the income deposited at a rate of 9 percent compounded continuously.

First, we need to calculate the integral:

\int_{0}^{16} 5500 e^{-0.02 t} e^{0.09(16-t)} dt

To evaluate this integral, we can use the following property of exponents:

e^{a+b} = e^a \cdot e^b

Using this property, we can rewrite the integral as:

\int_{0}^{16} 5500 e^{-0.02 t} e^{0.09(16-t)} dt = \int_{0}^{16} 5500 e^{-0.02 t + 1.44 - 0.09t} dt

Simplifying the exponent, we get:

\int_{0}^{16} 5500 e^{-0.11 t + 1.44} dt

Now, we can evaluate the integral:

\int_{0}^{16} 5500 e^{-0.11 t + 1.44} dt = 5500 \left[ \frac{e^{-0.11 t + 1.44}}{-0.11} \right]_{0}^{16}

Evaluating the expression, we get:

5500 \left[ \frac{e^{-0.11 t + 1.44}}{-0.11} \right]_{0}^{16} = 5500 \left[ \frac{e^{-1.76}}{-0.11} - \frac{e^{1.44}}{-0.11} \right]

Using a calculator to evaluate the expression, we get:

5500 \left[ \frac{e^{-1.76}}{-0.11} - \frac{e^{1.44}}{-0.11} \right] = 5500 \left[ -0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000<br/> # **Q&A: Finding the Value of an Income Stream after 16 Years**

Introduction

In our previous article, we explored the concept of finding the value of an income stream after a certain period of time. We used the given rate of flow of revenue, $R(t) = 5500 e^{-0.02 t}$ , and the income deposited at a rate of 9 percent compounded continuously to find the value of the income stream after 16 years. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the rate of flow of revenue, and how is it calculated?

A: The rate of flow of revenue, $R(t)$ , is a function that represents the rate at which revenue is generated over time. In this case, the rate of flow of revenue is given by the equation $R(t) = 5500 e^{-0.02 t}$ . This equation indicates that the rate of flow of revenue decreases exponentially over time, with a decay rate of 2% per year.

Q: What is continuous compounding, and how is it used in this problem?

A: Continuous compounding is a method of calculating interest on an investment where the interest is compounded continuously over time. This means that the interest is added to the principal at a rate that is proportional to the current balance. In this case, the income is deposited at a rate of 9 percent compounded continuously.

Q: How is the future value of the income stream calculated?

A: The future value of the income stream is calculated using the formula for continuous compounding: