The rate of a continuous money flow starts at $700 and increases exponentially at 4% per year for 20 years. Find the present value and accumulated amount if interest earned is 8% compounded continuously. The present value is $ (Do not round until the final answer. Then round to the nearest cent as needed.)
The rate of a continuous money flow starts at $700 and increases exponentially at 4% per year for 20 years. Find the present value and accumulated amount if interest earned is 8% compounded continuously. The present value is $ (Do not round until the final answer. Then round to the nearest cent as needed.)
Chapter9: Sequences, Probability And Counting Theory
Section: Chapter Questions
Problem 30RE: The twins Sarah and Scott both opened retirement accounts on their 2lst birthday. Sarah deposits...
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Question
![**Problem Statement:**
The rate of a continuous money flow starts at $700 and increases exponentially at 4% per year for 20 years. Find the present value and accumulated amount if the interest earned is 8% compounded continuously.
---
**Solution Format:**
*The present value is $ ____.*
*(Do not round until the final answer. Then round to the nearest cent as needed.)*
**Explanation:**
1. **Derive the Function for Money Flow:**
The rate \( R(t) \) of the continuous money flow can be represented as:
\[
R(t) = 700e^{0.04t}
\]
Here, \(700\) is the initial rate, \(0.04\) is the growth rate, and \(t\) represents time in years.
2. **Present Value Calculation:**
To find the present value \( PV \), we need to discount the money flow rate to present value using the formula:
\[
PV = \int_0^{20} R(t) e^{-0.08t} \, dt
\]
Substituting \( R(t) \) gives:
\[
PV = \int_0^{20} 700e^{0.04t} e^{-0.08t} \, dt = \int_0^{20} 700e^{-0.04t} \, dt
\]
3. **Solve the Integral:**
\[
PV = 700 \int_0^{20} e^{-0.04t} \, dt
\]
The indefinite integral of \( e^{-0.04t} \) is \( \frac{e^{-0.04t}}{-0.04} \):
\[
PV = 700 \left[ \frac{e^{-0.04t}}{-0.04} \right]_0^{20}
\]
Evaluating this from \( t = 0 \) to \( t = 20 \):
\[
PV = 700 \left( \frac{e^{-0.04 \cdot 20} - e^{0}}{-0.04} \right)
\]
Simplify the exponent:
\[
PV = 700 \left( \frac{e^{-0.8} - 1}{-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F756689fc-aa25-4045-a342-f553545a40f2%2F31588f0c-61a1-451b-911e-35cc6c682899%2Frqg9tmv_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
The rate of a continuous money flow starts at $700 and increases exponentially at 4% per year for 20 years. Find the present value and accumulated amount if the interest earned is 8% compounded continuously.
---
**Solution Format:**
*The present value is $ ____.*
*(Do not round until the final answer. Then round to the nearest cent as needed.)*
**Explanation:**
1. **Derive the Function for Money Flow:**
The rate \( R(t) \) of the continuous money flow can be represented as:
\[
R(t) = 700e^{0.04t}
\]
Here, \(700\) is the initial rate, \(0.04\) is the growth rate, and \(t\) represents time in years.
2. **Present Value Calculation:**
To find the present value \( PV \), we need to discount the money flow rate to present value using the formula:
\[
PV = \int_0^{20} R(t) e^{-0.08t} \, dt
\]
Substituting \( R(t) \) gives:
\[
PV = \int_0^{20} 700e^{0.04t} e^{-0.08t} \, dt = \int_0^{20} 700e^{-0.04t} \, dt
\]
3. **Solve the Integral:**
\[
PV = 700 \int_0^{20} e^{-0.04t} \, dt
\]
The indefinite integral of \( e^{-0.04t} \) is \( \frac{e^{-0.04t}}{-0.04} \):
\[
PV = 700 \left[ \frac{e^{-0.04t}}{-0.04} \right]_0^{20}
\]
Evaluating this from \( t = 0 \) to \( t = 20 \):
\[
PV = 700 \left( \frac{e^{-0.04 \cdot 20} - e^{0}}{-0.04} \right)
\]
Simplify the exponent:
\[
PV = 700 \left( \frac{e^{-0.8} - 1}{-
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