An investment produces a perpetual stream of income with a flow rate of R(t) = 1,500 e 0.02t. Find the capital value at an interest rate of 7% compounded continuously. The capital value is $
An investment produces a perpetual stream of income with a flow rate of R(t) = 1,500 e 0.02t. Find the capital value at an interest rate of 7% compounded continuously. The capital value is $
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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![**Problem Statement:**
An investment produces a perpetual stream of income with a flow rate of \( R(t) = 1,500 e^{0.02t} \). Find the capital value at an interest rate of 7% compounded continuously.
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**Solution:**
To find the capital value of the perpetual income stream, we can use the formula for the present value of a continuous income stream. Given the flow rate \( R(t) = 1,500 e^{0.02t} \) and an interest rate of 7% (or 0.07 in decimal form), the capital value \( C \) is computed using the integral:
\[ C = \int_{0}^{\infty} R(t) \cdot e^{-rt} \, dt \]
Substituting \( R(t) \) and \( r \):
\[ C = \int_{0}^{\infty} 1,500 e^{0.02t} \cdot e^{-0.07t} \, dt \]
\[ C = 1,500 \int_{0}^{\infty} e^{(0.02 - 0.07)t} \, dt \]
\[ C = 1,500 \int_{0}^{\infty} e^{-0.05t} \, dt \]
To solve the integral, use the formula for the integral of an exponential function:
\[ \int e^{at} \, dt = \frac{e^{at}}{a} + C \]
Applying this:
\[ C = 1,500 \left[ \frac{e^{-0.05t}}{-0.05} \right]_{0}^{\infty} \]
Evaluate the limits:
\[ C = 1,500 \left( \frac{e^{-0.05 \cdot \infty} - e^{-0.05 \cdot 0}}{-0.05} \right) \]
\[ C = 1,500 \left( \frac{0 - 1}{-0.05} \right) \]
\[ C = 1,500 \left( \frac{-1}{-0.05} \right) \]
\[ C = 1,500 \left( 20 \right) \]
\[ C = 30,000 \]
Therefore, the capital value is:
**The capital value is $](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F756689fc-aa25-4045-a342-f553545a40f2%2Fd723dd3d-70ce-49f5-ba3b-c50f210f91d1%2Fcxb2gne_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
An investment produces a perpetual stream of income with a flow rate of \( R(t) = 1,500 e^{0.02t} \). Find the capital value at an interest rate of 7% compounded continuously.
---
**Solution:**
To find the capital value of the perpetual income stream, we can use the formula for the present value of a continuous income stream. Given the flow rate \( R(t) = 1,500 e^{0.02t} \) and an interest rate of 7% (or 0.07 in decimal form), the capital value \( C \) is computed using the integral:
\[ C = \int_{0}^{\infty} R(t) \cdot e^{-rt} \, dt \]
Substituting \( R(t) \) and \( r \):
\[ C = \int_{0}^{\infty} 1,500 e^{0.02t} \cdot e^{-0.07t} \, dt \]
\[ C = 1,500 \int_{0}^{\infty} e^{(0.02 - 0.07)t} \, dt \]
\[ C = 1,500 \int_{0}^{\infty} e^{-0.05t} \, dt \]
To solve the integral, use the formula for the integral of an exponential function:
\[ \int e^{at} \, dt = \frac{e^{at}}{a} + C \]
Applying this:
\[ C = 1,500 \left[ \frac{e^{-0.05t}}{-0.05} \right]_{0}^{\infty} \]
Evaluate the limits:
\[ C = 1,500 \left( \frac{e^{-0.05 \cdot \infty} - e^{-0.05 \cdot 0}}{-0.05} \right) \]
\[ C = 1,500 \left( \frac{0 - 1}{-0.05} \right) \]
\[ C = 1,500 \left( \frac{-1}{-0.05} \right) \]
\[ C = 1,500 \left( 20 \right) \]
\[ C = 30,000 \]
Therefore, the capital value is:
**The capital value is $
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