Brandon invested $41,000 in an account paying an interest rate of 3 % compounded continuously. Evan invested $41,000 in an account paying an interest rate of 3% compounded quarterly. To the nearest dollar, how much money would Brandon have in his account when Evan's money has doubled in value?
Brandon invested $41,000 in an account paying an interest rate of 3 % compounded continuously. Evan invested $41,000 in an account paying an interest rate of 3% compounded quarterly. To the nearest dollar, how much money would Brandon have in his account when Evan's money has doubled in value?
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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![### Investment Problem
#### Problem Statement:
**Date/Time:** Jun 14, 3:09:36 PM
Brandon invested $41,000 in an account paying an interest rate of \(3\frac{3}{4}\)% compounded continuously. Evan invested $41,000 in an account paying an interest rate of \(3\frac{7}{8}\)% compounded quarterly. To the **nearest dollar**, how much money would Brandon have in his account when Evan's money has doubled in value?
**Answer:**
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**Note:** You have 2 attempts to answer this question.
#### Explanation:
In this problem, we are comparing two investment strategies with different compounding methods and interest rates. The goal is to determine the future value of Brandon's investment when Evan's investment has doubled.
- **Brandon's Investment:**
- Initial Amount: $41,000
- Interest Rate: \(3\frac{3}{4}\)% compounded continuously
- **Evan's Investment:**
- Initial Amount: $41,000
- Interest Rate: \(3\frac{7}{8}\)% compounded quarterly
The key formulae required to solve this problem involve the compound interest formula for different compounding intervals:
1. **Continuous Compounding:**
\[ A = P \cdot e^{rt} \]
2. **Quarterly Compounding:**
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \(A\) is the amount of money accumulated after \(n\) periods, including interest.
- \(P\) is the principal amount (the initial sum of money).
- \(r\) is the annual interest rate (decimal).
- \(t\) is the time the money is invested for.
- \(n\) is the number of times that interest is compounded per year.
In this case, you will have to:
1. Calculate the time \(t\) it takes for Evan's investment to double using the quarterly compounding formula.
2. Use that time \(t\) to find the value of Brandon's investment using the continuous compounding formula.
This problem tests your understanding of the compounding interest concepts and requires careful calculations to find the solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2c289932-425f-49ed-b510-3fb76f7fe4c9%2F457275fc-9360-4739-bdf2-87500290e2fd%2Fdlisg5r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Investment Problem
#### Problem Statement:
**Date/Time:** Jun 14, 3:09:36 PM
Brandon invested $41,000 in an account paying an interest rate of \(3\frac{3}{4}\)% compounded continuously. Evan invested $41,000 in an account paying an interest rate of \(3\frac{7}{8}\)% compounded quarterly. To the **nearest dollar**, how much money would Brandon have in his account when Evan's money has doubled in value?
**Answer:**
[Text box for answer input]
**Submit Answer Button**
**Note:** You have 2 attempts to answer this question.
#### Explanation:
In this problem, we are comparing two investment strategies with different compounding methods and interest rates. The goal is to determine the future value of Brandon's investment when Evan's investment has doubled.
- **Brandon's Investment:**
- Initial Amount: $41,000
- Interest Rate: \(3\frac{3}{4}\)% compounded continuously
- **Evan's Investment:**
- Initial Amount: $41,000
- Interest Rate: \(3\frac{7}{8}\)% compounded quarterly
The key formulae required to solve this problem involve the compound interest formula for different compounding intervals:
1. **Continuous Compounding:**
\[ A = P \cdot e^{rt} \]
2. **Quarterly Compounding:**
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \(A\) is the amount of money accumulated after \(n\) periods, including interest.
- \(P\) is the principal amount (the initial sum of money).
- \(r\) is the annual interest rate (decimal).
- \(t\) is the time the money is invested for.
- \(n\) is the number of times that interest is compounded per year.
In this case, you will have to:
1. Calculate the time \(t\) it takes for Evan's investment to double using the quarterly compounding formula.
2. Use that time \(t\) to find the value of Brandon's investment using the continuous compounding formula.
This problem tests your understanding of the compounding interest concepts and requires careful calculations to find the solution.
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