Find the present value of a continuous stream of income over 3 years when the rate of income is constant at $31,000 per year and the interest rate is 3% The present value is $. (Round to the nearest dollar as needed.)
Find the present value of a continuous stream of income over 3 years when the rate of income is constant at $31,000 per year and the interest rate is 3% The present value is $. (Round to the nearest dollar as needed.)
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter7: Exponents And Exponential Functions
Section7.8: Transforming Exponential Expressions
Problem 1GP
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![### Calculating the Present Value of a Continuous Stream of Income
Consider a continuous stream of income over a period of 3 years, wherein the rate of income is constant at $31,000 per year, and the interest rate is 3%.
The problem is stated as follows:
> **Find the present value of a continuous stream of income over 3 years when the rate of income is constant at $31,000 per year and the interest rate is 3%.**
Below this statement, there is a calculation interface to enter the value:
> **The present value is \$ ⬜.** *(Round to the nearest dollar as needed.)*
#### Explanation
To find the present value (\(PV\)) of the continuous stream of income with a constant rate, we use the formula:
\[ PV = \int_{0}^{T} R(t) e^{-rt} \, dt \]
Where:
- \( R(t) \) is the rate of income per year.
- \( r \) is the interest rate.
- \( T \) is the time period over which the income is received.
Given the information:
- \( R(t) = 31,000 \)
- \( r = 0.03 \)
- \( T = 3 \)
Plugging in these values:
\[ PV = \int_{0}^{3} 31,000 e^{-0.03t} \, dt \]
This integral can be solved to find the exact present value of this income stream, which would then be rounded to the nearest dollar.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c1127c9-3767-40bf-bfb8-794124c4dac3%2F7d826981-190f-490d-9db7-ebe27a93b25b%2Fg61lgpo_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Present Value of a Continuous Stream of Income
Consider a continuous stream of income over a period of 3 years, wherein the rate of income is constant at $31,000 per year, and the interest rate is 3%.
The problem is stated as follows:
> **Find the present value of a continuous stream of income over 3 years when the rate of income is constant at $31,000 per year and the interest rate is 3%.**
Below this statement, there is a calculation interface to enter the value:
> **The present value is \$ ⬜.** *(Round to the nearest dollar as needed.)*
#### Explanation
To find the present value (\(PV\)) of the continuous stream of income with a constant rate, we use the formula:
\[ PV = \int_{0}^{T} R(t) e^{-rt} \, dt \]
Where:
- \( R(t) \) is the rate of income per year.
- \( r \) is the interest rate.
- \( T \) is the time period over which the income is received.
Given the information:
- \( R(t) = 31,000 \)
- \( r = 0.03 \)
- \( T = 3 \)
Plugging in these values:
\[ PV = \int_{0}^{3} 31,000 e^{-0.03t} \, dt \]
This integral can be solved to find the exact present value of this income stream, which would then be rounded to the nearest dollar.
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