Find the present value of a continuous stream of income over 4 years when the rate of income is constant at $34,000 per year and the interest rate is 8%.
Find the present value of a continuous stream of income over 4 years when the rate of income is constant at $34,000 per year and the interest rate is 8%.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Calculating the Present Value of a Continuous Stream of Income
#### Problem Statement:
Find the present value of a continuous stream of income over 4 years when the rate of income is constant at $34,000 per year and the interest rate is 8%.
#### Detailed Solution:
To solve for the present value (PV) of a continuous stream of income, we use the following formula for the present value of continuous income:
\[ PV = \int_{0}^{T} R(t) e^{-rt} dt \]
Since the income rate \( R(t) \) is constant at $34,000 per year, the formula simplifies to:
\[ PV = R \int_{0}^{T} e^{-rt} dt \]
where:
- \( R \) is the annual income rate ($34,000/year)
- \( r \) is the annual interest rate (8% or 0.08)
- \( T \) is the total period (4 years)
The integral simplifies to:
\[ PV = 34{,}000 \int_{0}^{4} e^{-0.08t} dt \]
Evaluating the integral:
\[ \int_{0}^{4} e^{-0.08t} dt \]
This is a standard exponential integral and can be solved as:
\[ \int e^{-0.08t} dt = \frac{-1}{0.08} e^{-0.08t} \]
Substituting the bounds from 0 to 4:
\[ \left[ \frac{-1}{0.08} e^{-0.08t} \right]_{0}^{4} \]
\[ = \frac{-1}{0.08} \left( e^{-0.32} - 1 \right) \]
\[ = \frac{-1}{0.08} (0.7261 - 1) \]
\[ = \frac{-1}{0.08} (-0.2739) \]
\[ = \frac{0.2739}{0.08} \]
\[ \approx 3.42375 \]
Then, multiplying by the income rate:
\[ PV = 34{,}000 \times 3.42375 \]
\[ \approx \$116{,}407 \]
#### Conclusion:
The present value is approximately **$116,407** (Rounded to the nearest dollar as needed).
Note: The specific integral values can be](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8558f12-eb43-4f23-8df4-76c708dcf941%2Fa3f4cd44-5231-4bcb-859b-7fdd8e9331cb%2Fc13t8c_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Present Value of a Continuous Stream of Income
#### Problem Statement:
Find the present value of a continuous stream of income over 4 years when the rate of income is constant at $34,000 per year and the interest rate is 8%.
#### Detailed Solution:
To solve for the present value (PV) of a continuous stream of income, we use the following formula for the present value of continuous income:
\[ PV = \int_{0}^{T} R(t) e^{-rt} dt \]
Since the income rate \( R(t) \) is constant at $34,000 per year, the formula simplifies to:
\[ PV = R \int_{0}^{T} e^{-rt} dt \]
where:
- \( R \) is the annual income rate ($34,000/year)
- \( r \) is the annual interest rate (8% or 0.08)
- \( T \) is the total period (4 years)
The integral simplifies to:
\[ PV = 34{,}000 \int_{0}^{4} e^{-0.08t} dt \]
Evaluating the integral:
\[ \int_{0}^{4} e^{-0.08t} dt \]
This is a standard exponential integral and can be solved as:
\[ \int e^{-0.08t} dt = \frac{-1}{0.08} e^{-0.08t} \]
Substituting the bounds from 0 to 4:
\[ \left[ \frac{-1}{0.08} e^{-0.08t} \right]_{0}^{4} \]
\[ = \frac{-1}{0.08} \left( e^{-0.32} - 1 \right) \]
\[ = \frac{-1}{0.08} (0.7261 - 1) \]
\[ = \frac{-1}{0.08} (-0.2739) \]
\[ = \frac{0.2739}{0.08} \]
\[ \approx 3.42375 \]
Then, multiplying by the income rate:
\[ PV = 34{,}000 \times 3.42375 \]
\[ \approx \$116{,}407 \]
#### Conclusion:
The present value is approximately **$116,407** (Rounded to the nearest dollar as needed).
Note: The specific integral values can be
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