Find the present value of a continuous stream of income over 5 years when the rate of income is constant at $33,000 per year and the interest rate is 8%.
Find the present value of a continuous stream of income over 5 years when the rate of income is constant at $33,000 per year and the interest rate is 8%.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 16EQ
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![### Present Value of Continuous Income Stream
In this exercise, we are tasked with finding the present value of a continuous stream of income over 5 years. The conditions given are:
- The rate of income is constant at $33,000 per year.
- The interest rate is 8%.
To determine the present value (PV) of this continuous income stream, the formula used is:
\[PV = \int_{0}^{T} R \cdot e^{-rt} \, dt\]
Where:
- \(R\) is the rate of income per year.
- \(r\) is the interest rate.
- \(T\) is the duration in years.
- \(t\) is the time variable.
Given:
- \(R = 33000 \text{ USD/year}\)
- \(r = 0.08\)
- \(T = 5 \text{ years}\)
Substituting these values into the formula, we get:
\[PV = \int_{0}^{5} 33000 \cdot e^{-0.08t} \, dt\]
To solve this integral:
1. Integrate the function \(\int e^{-0.08t} \, dt\).
2. Apply the limits from 0 to 5.
After evaluating the integral, the present value should be rounded to the nearest dollar as needed.
**Note**: In an actual educational setting, students would be guided to solve this integral step-by-step either manually or using computational tools for exact evaluation.
**Result Placeholder**:
The present value is \( \$ \boxed{} \).
**Instruction**: (Round to the nearest dollar as needed.)
This exercise helps students understand key financial concepts and the application of calculus in economic scenarios.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8558f12-eb43-4f23-8df4-76c708dcf941%2Ffee06945-b5cc-47a0-83ea-4f71cbf59faf%2Fqnfpbvf_processed.png&w=3840&q=75)
Transcribed Image Text:### Present Value of Continuous Income Stream
In this exercise, we are tasked with finding the present value of a continuous stream of income over 5 years. The conditions given are:
- The rate of income is constant at $33,000 per year.
- The interest rate is 8%.
To determine the present value (PV) of this continuous income stream, the formula used is:
\[PV = \int_{0}^{T} R \cdot e^{-rt} \, dt\]
Where:
- \(R\) is the rate of income per year.
- \(r\) is the interest rate.
- \(T\) is the duration in years.
- \(t\) is the time variable.
Given:
- \(R = 33000 \text{ USD/year}\)
- \(r = 0.08\)
- \(T = 5 \text{ years}\)
Substituting these values into the formula, we get:
\[PV = \int_{0}^{5} 33000 \cdot e^{-0.08t} \, dt\]
To solve this integral:
1. Integrate the function \(\int e^{-0.08t} \, dt\).
2. Apply the limits from 0 to 5.
After evaluating the integral, the present value should be rounded to the nearest dollar as needed.
**Note**: In an actual educational setting, students would be guided to solve this integral step-by-step either manually or using computational tools for exact evaluation.
**Result Placeholder**:
The present value is \( \$ \boxed{} \).
**Instruction**: (Round to the nearest dollar as needed.)
This exercise helps students understand key financial concepts and the application of calculus in economic scenarios.
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