Teri invested $1500 in an account with an interest rate of 2.25% compounded continuously. How long will it take for Teri's account to earn $5000?
Teri invested $1500 in an account with an interest rate of 2.25% compounded continuously. How long will it take for Teri's account to earn $5000?
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...
Related questions
Question
![**Problem Statement: Continuous Compound Interest Calculation**
Teri invested $1500 in an account with an interest rate of 2.25% compounded continuously. How long will it take for Teri's account to earn $5000?
**Instructions: Show all steps.**
---
**Solution Steps:**
To solve this problem, we use the formula for continuous compound interest:
\[ A = Pe^{rt} \]
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 rate of interest per year (as a decimal).
- \( t \) is the time the money is invested for in years.
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
### Step-by-step Solution:
1. **Identify the values:**
- \( P = 1500 \) (initial investment)
- \( r = 0.0225 \) (annual interest rate in decimal form)
- \( A = 5000 \) (final amount)
2. **Plug the values into the formula:**
\[ 5000 = 1500 \times e^{0.0225t} \]
3. **Solve for \( t \):**
- Divide both sides by 1500:
\[ \frac{5000}{1500} = e^{0.0225t} \]
\[ \frac{500}{150} = e^{0.0225t} \]
\[ \frac{50}{15} \approx 3.3333 = e^{0.0225t} \]
- Take the natural logarithm of both sides:
\[ \ln(3.3333) = 0.0225t \]
- Calculate \( \ln(3.3333) \):
\[ \ln(3.3333) \approx 1.2039 \]
- Solve for \( t \):
\[ t = \frac{1.2039}{0.0225} \]
- Calculate \( t \):
\[ t \approx 53.51 \]
### Conclusion:
It will take approximately 53.51 years for Teri's account to grow to $5000 with an](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fc5117c-6b63-40df-86f1-94f8b6471697%2Fa896276f-4232-446e-9157-a96b4369e6e1%2Fun4xey_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Continuous Compound Interest Calculation**
Teri invested $1500 in an account with an interest rate of 2.25% compounded continuously. How long will it take for Teri's account to earn $5000?
**Instructions: Show all steps.**
---
**Solution Steps:**
To solve this problem, we use the formula for continuous compound interest:
\[ A = Pe^{rt} \]
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 rate of interest per year (as a decimal).
- \( t \) is the time the money is invested for in years.
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
### Step-by-step Solution:
1. **Identify the values:**
- \( P = 1500 \) (initial investment)
- \( r = 0.0225 \) (annual interest rate in decimal form)
- \( A = 5000 \) (final amount)
2. **Plug the values into the formula:**
\[ 5000 = 1500 \times e^{0.0225t} \]
3. **Solve for \( t \):**
- Divide both sides by 1500:
\[ \frac{5000}{1500} = e^{0.0225t} \]
\[ \frac{500}{150} = e^{0.0225t} \]
\[ \frac{50}{15} \approx 3.3333 = e^{0.0225t} \]
- Take the natural logarithm of both sides:
\[ \ln(3.3333) = 0.0225t \]
- Calculate \( \ln(3.3333) \):
\[ \ln(3.3333) \approx 1.2039 \]
- Solve for \( t \):
\[ t = \frac{1.2039}{0.0225} \]
- Calculate \( t \):
\[ t \approx 53.51 \]
### Conclusion:
It will take approximately 53.51 years for Teri's account to grow to $5000 with an
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning