Find the accumulated future value of the continuous income stream at rate R(t), for the given time T, and interest rate k, compounded continuously. R(t) = $70,000, T = 18 years, k = 5% The accumulated future value is $. (Round to the nearest ten dollars as needed.)
Find the accumulated future value of the continuous income stream at rate R(t), for the given time T, and interest rate k, compounded continuously. R(t) = $70,000, T = 18 years, k = 5% The accumulated future value is $. (Round to the nearest ten dollars as needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Accumulated Future Value of Continuous Income Stream
To find the accumulated future value (AFV) of a continuous income stream at rate \( R(t) \), for a given time \( T \) and interest rate \( k \), compounded continuously, consider the following parameters:
\[
R(t) = \$70,000, \quad T = 18 \text{ years}, \quad k = 5\%
\]
---
**Calculation:**
The accumulated future value is calculated using the formula for continuous compounding.
\[ AFV = \int_{0}^{T} R(t) e^{k(T-t)} \, dt \]
Given that \( R(t) \) is constant at $70,000, the formula simplifies to:
\[ AFV = R \int_{0}^{T} e^{k(T-t)} \, dt \]
Finally, we compute the definite integral to find the accumulated future value.
The accumulated future value is $________________ (Round to the nearest ten dollars as needed.)
---
Please use the above parameters to perform your calculations for determining the AFV.
**Note:** The AFV formula and integral should be computed accordingly to yield the required future value.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F281e2888-f9a2-4618-9a6a-25b82d3b31a2%2Fd4e437b8-f943-4aa5-8360-84269a35bc4f%2Fnwc75r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Accumulated Future Value of Continuous Income Stream
To find the accumulated future value (AFV) of a continuous income stream at rate \( R(t) \), for a given time \( T \) and interest rate \( k \), compounded continuously, consider the following parameters:
\[
R(t) = \$70,000, \quad T = 18 \text{ years}, \quad k = 5\%
\]
---
**Calculation:**
The accumulated future value is calculated using the formula for continuous compounding.
\[ AFV = \int_{0}^{T} R(t) e^{k(T-t)} \, dt \]
Given that \( R(t) \) is constant at $70,000, the formula simplifies to:
\[ AFV = R \int_{0}^{T} e^{k(T-t)} \, dt \]
Finally, we compute the definite integral to find the accumulated future value.
The accumulated future value is $________________ (Round to the nearest ten dollars as needed.)
---
Please use the above parameters to perform your calculations for determining the AFV.
**Note:** The AFV formula and integral should be computed accordingly to yield the required future value.
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