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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Question
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![Here is the transcription of the text from the image, formatted for an educational website:
---
### Compound Interest Problem
**Problem Statement**:
Ten thousand dollars is invested at 9/6/15 interest compounded continuously.
**Question**:
When will the investment be worth $41,787?
---
To solve this problem, you need to understand the formula for continuous compounding interest, which is given by:
\[ A = P e^{rt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( t \) is the time the money is invested for, in years.
- \( e \) is Euler's number (approximately equal to 2.71828).
Given data:
- \( P = 10,000 \) dollars
- \( A = 41,787 \) dollars
- Convert the interest rate from 9/6/15 to decimal format and solve for \( t \).
---
**Steps to Solve**:
1. Convert the annual interest rate to decimal form.
2. Substitute the values into the continuous compound interest formula.
3. Solve for \( t \) using logarithmic functions.
---
By following these steps, you can determine how long it will take for the investment to grow to $41,787.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b8d1efa-c07b-4629-8963-670ddc0d1c0b%2Feabff544-e4e8-4211-9bc5-633ebbd6cc47%2Fh1dyz2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Here is the transcription of the text from the image, formatted for an educational website:
---
### Compound Interest Problem
**Problem Statement**:
Ten thousand dollars is invested at 9/6/15 interest compounded continuously.
**Question**:
When will the investment be worth $41,787?
---
To solve this problem, you need to understand the formula for continuous compounding interest, which is given by:
\[ A = P e^{rt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( t \) is the time the money is invested for, in years.
- \( e \) is Euler's number (approximately equal to 2.71828).
Given data:
- \( P = 10,000 \) dollars
- \( A = 41,787 \) dollars
- Convert the interest rate from 9/6/15 to decimal format and solve for \( t \).
---
**Steps to Solve**:
1. Convert the annual interest rate to decimal form.
2. Substitute the values into the continuous compound interest formula.
3. Solve for \( t \) using logarithmic functions.
---
By following these steps, you can determine how long it will take for the investment to grow to $41,787.
![**Sample: Compound Continuously**
In this illustration, we focus on the formula for continuous compounding. Continuous compounding refers to the mathematical limit that compound interest can reach if it’s calculated and re-invested into an account's balance over an infinite number of times per period.
The formula used is:
\[ A(t) = Pe^{rt} \]
Where:
- **A(t)** is the amount of money accumulated after time \( t \) (includes principal and interest).
- **P** is the principal amount (the initial amount of money).
- **e** is the base of the natural logarithm, approximately equal to 2.71828.
- **r** is the annual interest rate (decimal).
- **t** is the time the money is invested or borrowed for, in years.
**Explanation of the Variables:**
1. **A(t)**: This represents the total amount after compounding over time.
2. **P**: This is the principal, the initial sum of money.
3. **r**: This is the rate of interest per year as a decimal. For instance, an interest rate of 5% would be \( 0.05 \).
4. **t**: This is the time the money is invested or borrowed for, expressed in years.
5. **e**: This mathematical constant is the base of the natural logarithm and is approximately equal to 2.71828.
This formula shows the exponential growth of the investment, as the interest is compounded continuously.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b8d1efa-c07b-4629-8963-670ddc0d1c0b%2Feabff544-e4e8-4211-9bc5-633ebbd6cc47%2Fagjm2ut_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Sample: Compound Continuously**
In this illustration, we focus on the formula for continuous compounding. Continuous compounding refers to the mathematical limit that compound interest can reach if it’s calculated and re-invested into an account's balance over an infinite number of times per period.
The formula used is:
\[ A(t) = Pe^{rt} \]
Where:
- **A(t)** is the amount of money accumulated after time \( t \) (includes principal and interest).
- **P** is the principal amount (the initial amount of money).
- **e** is the base of the natural logarithm, approximately equal to 2.71828.
- **r** is the annual interest rate (decimal).
- **t** is the time the money is invested or borrowed for, in years.
**Explanation of the Variables:**
1. **A(t)**: This represents the total amount after compounding over time.
2. **P**: This is the principal, the initial sum of money.
3. **r**: This is the rate of interest per year as a decimal. For instance, an interest rate of 5% would be \( 0.05 \).
4. **t**: This is the time the money is invested or borrowed for, expressed in years.
5. **e**: This mathematical constant is the base of the natural logarithm and is approximately equal to 2.71828.
This formula shows the exponential growth of the investment, as the interest is compounded continuously.
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