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 5:** Determine the time it takes an amount of money to triple if compounded continuously at an annual rate of 12%.
*Explanation:*
To solve this problem, you would use the formula for continuous compounding, which is based on the equation:
\[ A = Pe^{rt} \]
Where:
- \( A \) is the amount of money after time \( t \).
- \( P \) is the principal amount (initial amount).
- \( r \) is the annual interest rate (as a decimal).
- \( t \) is the time in years.
- \( e \) is the base of the natural logarithm.
Since the goal is for the money to triple, we have \( A = 3P \). Substituting into the formula gives us:
\[ 3P = Pe^{0.12t} \]
By dividing both sides by \( P \) (assuming \( P \neq 0 \)), we get:
\[ 3 = e^{0.12t} \]
Taking the natural logarithm of both sides:
\[ \ln(3) = 0.12t \]
Solving for \( t \):
\[ t = \frac{\ln(3)}{0.12} \]
This result gives you the time it takes for the money to triple. You can calculate this using a calculator to find the exact number of years.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4eaca33a-bcfa-4336-aecf-f320ab80f63b%2Fdb6b617f-bb00-42b3-99b2-8c724f6d8377%2Fce40bf_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 5:** Determine the time it takes an amount of money to triple if compounded continuously at an annual rate of 12%.
*Explanation:*
To solve this problem, you would use the formula for continuous compounding, which is based on the equation:
\[ A = Pe^{rt} \]
Where:
- \( A \) is the amount of money after time \( t \).
- \( P \) is the principal amount (initial amount).
- \( r \) is the annual interest rate (as a decimal).
- \( t \) is the time in years.
- \( e \) is the base of the natural logarithm.
Since the goal is for the money to triple, we have \( A = 3P \). Substituting into the formula gives us:
\[ 3P = Pe^{0.12t} \]
By dividing both sides by \( P \) (assuming \( P \neq 0 \)), we get:
\[ 3 = e^{0.12t} \]
Taking the natural logarithm of both sides:
\[ \ln(3) = 0.12t \]
Solving for \( t \):
\[ t = \frac{\ln(3)}{0.12} \]
This result gives you the time it takes for the money to triple. You can calculate this using a calculator to find the exact number of years.
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 3 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