Upon the death of his aunt, Lucien receives an inheritance of $40,000, which he invests for 13 years at 6.4%, compounded continuously. What is the future value of the inheritance? ***
Upon the death of his aunt, Lucien receives an inheritance of $40,000, which he invests for 13 years at 6.4%, compounded continuously. What is the future value of the inheritance? ***
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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![**Future Value of Inheritance Invested at Continuous Compound Interest**
**Problem Statement:**
Upon the death of his aunt, Lucien receives an inheritance of $40,000, which he invests for 13 years at 6.4%, compounded continuously. What is the future value of the inheritance?
---
**Formula and Solution:**
To find the future value (\(A\)) of an investment compounded continuously, we use the formula:
\[ A = P \cdot e^{rt} \]
Where:
- \( P \) is the principal amount (initial investment), which is $40,000.
- \( r \) is the annual interest rate, expressed as a decimal, which is 6.4%, or 0.064.
- \( t \) is the time the money is invested for, in years, which is 13 years.
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
Let's plug in the values and calculate:
\[ A = 40,000 \cdot e^{(0.064 \cdot 13)} \]
**Computation:**
1. Calculate the exponent:
\[ 0.064 \cdot 13 = 0.832 \]
2. Find \( e^{0.832} \):
Using a scientific calculator or online computation tool:
\[ e^{0.832} \approx 2.29857 \]
3. Multiply the principal by this value:
\[ 40,000 \cdot 2.29857 \approx 91,942.80 \]
So, the future value is approximately $91,942.80.
---
**Conclusion:**
The future value is **$91,942.80**.
**Note:** Remember to round to the nearest cent as needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8080b8ef-a292-427a-a45d-f2d23639cb67%2Faa6eabde-3843-40ff-ab80-33e6b0aa7457%2F7ei7mqf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Future Value of Inheritance Invested at Continuous Compound Interest**
**Problem Statement:**
Upon the death of his aunt, Lucien receives an inheritance of $40,000, which he invests for 13 years at 6.4%, compounded continuously. What is the future value of the inheritance?
---
**Formula and Solution:**
To find the future value (\(A\)) of an investment compounded continuously, we use the formula:
\[ A = P \cdot e^{rt} \]
Where:
- \( P \) is the principal amount (initial investment), which is $40,000.
- \( r \) is the annual interest rate, expressed as a decimal, which is 6.4%, or 0.064.
- \( t \) is the time the money is invested for, in years, which is 13 years.
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
Let's plug in the values and calculate:
\[ A = 40,000 \cdot e^{(0.064 \cdot 13)} \]
**Computation:**
1. Calculate the exponent:
\[ 0.064 \cdot 13 = 0.832 \]
2. Find \( e^{0.832} \):
Using a scientific calculator or online computation tool:
\[ e^{0.832} \approx 2.29857 \]
3. Multiply the principal by this value:
\[ 40,000 \cdot 2.29857 \approx 91,942.80 \]
So, the future value is approximately $91,942.80.
---
**Conclusion:**
The future value is **$91,942.80**.
**Note:** Remember to round to the nearest cent as needed.
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