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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![**Compound Interest Calculation**
**Problem Statement:**
If $15,000 is invested at 3% compounded quarterly, what is the amount after 7 years?
**Solution:**
To find the future value of the investment, we use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount ($15,000).
- \( r \) is the annual interest rate (0.03).
- \( n \) is the number of times that interest is compounded per year (4 for quarterly).
- \( t \) is the time the money is invested or borrowed for, in years (7).
Thus, substituting the given values:
\[ A = 15000 \left(1 + \frac{0.03}{4}\right)^{4 \times 7} \]
This simplifies to:
\[ A = 15000 \left(1 + 0.0075\right)^{28} \]
\[ A = 15000 \times (1.0075)^{28} \]
Using a calculator, the resulting amount \( A \) is:
\[ A = \$18490.68 \]
The investment will grow to $18,490.68 after 7 years at an interest rate of 3% compounded quarterly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29b25716-5eb6-4537-b644-b049114aff43%2F539f9df8-8d7e-446e-87d9-e9ca5b672e1c%2F2j6fu68_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Compound Interest Calculation**
**Problem Statement:**
If $15,000 is invested at 3% compounded quarterly, what is the amount after 7 years?
**Solution:**
To find the future value of the investment, we use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount ($15,000).
- \( r \) is the annual interest rate (0.03).
- \( n \) is the number of times that interest is compounded per year (4 for quarterly).
- \( t \) is the time the money is invested or borrowed for, in years (7).
Thus, substituting the given values:
\[ A = 15000 \left(1 + \frac{0.03}{4}\right)^{4 \times 7} \]
This simplifies to:
\[ A = 15000 \left(1 + 0.0075\right)^{28} \]
\[ A = 15000 \times (1.0075)^{28} \]
Using a calculator, the resulting amount \( A \) is:
\[ A = \$18490.68 \]
The investment will grow to $18,490.68 after 7 years at an interest rate of 3% compounded quarterly.
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